Calculus Examples

$f (x) = 3 x^{4} + 4 x^{3} - 6 x^{2}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $3 x^{4} + 4 x^{3} - 6 x^{2}$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 6 x^{2}]$ .

$f' (x) = \frac{d}{d x} (3 x^{4}) + \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 6 x^{2})$

Step 1.1.2

Evaluate $\frac{d}{d x} [3 x^{4}]$ .

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Step 1.1.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{4}]$ .

$f' (x) = 3 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 6 x^{2})$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f' (x) = 3 (4 x^{3}) + \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 6 x^{2})$

Step 1.1.2.3

Multiply $4$ by $3$ .

$f' (x) = 12 x^{3} + \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 6 x^{2})$

Step 1.1.3

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 1.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$f' (x) = 12 x^{3} + 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 6 x^{2})$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f' (x) = 12 x^{3} + 4 (3 x^{2}) + \frac{d}{d x} (- 6 x^{2})$

Step 1.1.3.3

Multiply $3$ by $4$ .

$f' (x) = 12 x^{3} + 12 x^{2} + \frac{d}{d x} (- 6 x^{2})$

Step 1.1.4

Evaluate $\frac{d}{d x} [- 6 x^{2}]$ .

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Step 1.1.4.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{2}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 12 x^{3} + 12 x^{2} - 6 \frac{d}{d x} x^{2}$

Step 1.1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f' (x) = 12 x^{3} + 12 x^{2} - 6 (2 x)$

Step 1.1.4.3

Multiply $2$ by $- 6$ .

$f' (x) = 12 x^{3} + 12 x^{2} - 12 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $12 x^{3} + 12 x^{2} - 12 x$ .

$12 x^{3} + 12 x^{2} - 12 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $12 x^{3} + 12 x^{2} - 12 x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$12 x^{3} + 12 x^{2} - 12 x = 0$

Step 2.2

Factor $12 x$ out of $12 x^{3} + 12 x^{2} - 12 x$ .

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Step 2.2.1

Factor $12 x$ out of $12 x^{3}$ .

$12 x (x^{2}) + 12 x^{2} - 12 x = 0$

Step 2.2.2

Factor $12 x$ out of $12 x^{2}$ .

$12 x (x^{2}) + 12 x (x) - 12 x = 0$

Step 2.2.3

Factor $12 x$ out of $- 12 x$ .

$12 x (x^{2}) + 12 x (x) + 12 x (- 1) = 0$

Step 2.2.4

Factor $12 x$ out of $12 x (x^{2}) + 12 x (x)$ .

$12 x (x^{2} + x) + 12 x (- 1) = 0$

Step 2.2.5

Factor $12 x$ out of $12 x (x^{2} + x) + 12 x (- 1)$ .

$12 x (x^{2} + x - 1) = 0$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x^{2} + x - 1 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $x^{2} + x - 1$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x^{2} + x - 1$ equal to $0$ .

$x^{2} + x - 1 = 0$

Step 2.5.2

Solve $x^{2} + x - 1 = 0$ for $x$ .

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Step 2.5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.5.2.2

Substitute the values $a = 1$ , $b = 1$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot - 1)}}{2 \cdot 1}$

Step 2.5.2.3

Simplify.

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Step 2.5.2.3.1

Simplify the numerator.

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Step 2.5.2.3.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 2.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 2.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 1}}{2 \cdot 1}$

Step 2.5.2.3.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{1 + 4}}{2 \cdot 1}$

Step 2.5.2.3.1.3

Add $1$ and $4$ .

$x = \frac{- 1 \pm \sqrt{5}}{2 \cdot 1}$

Step 2.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{5}}{2}$

Step 2.5.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.5.2.4.1

Simplify the numerator.

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Step 2.5.2.4.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 2.5.2.4.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 2.5.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 1}}{2 \cdot 1}$

Step 2.5.2.4.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{1 + 4}}{2 \cdot 1}$

Step 2.5.2.4.1.3

Add $1$ and $4$ .

$x = \frac{- 1 \pm \sqrt{5}}{2 \cdot 1}$

Step 2.5.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{5}}{2}$

Step 2.5.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{5}}{2}$

Step 2.5.2.4.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 + \sqrt{5}}{2}$

Step 2.5.2.4.5

Factor $- 1$ out of $\sqrt{5}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{5})}{2}$

Step 2.5.2.4.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{5})$ .

$x = \frac{- 1 (1 - \sqrt{5})}{2}$

Step 2.5.2.4.7

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{5}}{2}$

Step 2.5.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.5.2.5.1

Simplify the numerator.

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Step 2.5.2.5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 2.5.2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 2.5.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 1}}{2 \cdot 1}$

Step 2.5.2.5.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{1 + 4}}{2 \cdot 1}$

Step 2.5.2.5.1.3

Add $1$ and $4$ .

$x = \frac{- 1 \pm \sqrt{5}}{2 \cdot 1}$

Step 2.5.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{5}}{2}$

Step 2.5.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{5}}{2}$

Step 2.5.2.5.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 - \sqrt{5}}{2}$

Step 2.5.2.5.5

Factor $- 1$ out of $- \sqrt{5}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{5})}{2}$

Step 2.5.2.5.6

Factor $- 1$ out of $- 1 (1) - (\sqrt{5})$ .

$x = \frac{- 1 (1 + \sqrt{5})}{2}$

Step 2.5.2.5.7

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{5}}{2}$

Step 2.5.2.6

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{5}}{2}, - \frac{1 + \sqrt{5}}{2}$

Step 2.6

The final solution is all the values that make $12 x (x^{2} + x - 1) = 0$ true.

$x = 0, - \frac{1 - \sqrt{5}}{2}, - \frac{1 + \sqrt{5}}{2}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $3 x^{4} + 4 x^{3} - 6 x^{2}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$3 {(0)}^{4} + 4 {(0)}^{3} - 6 {(0)}^{2}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Raising $0$ to any positive power yields $0$ .

$3 \cdot 0 + 4 {(0)}^{3} - 6 {(0)}^{2}$

Step 4.1.2.1.2

Multiply $3$ by $0$ .

$0 + 4 {(0)}^{3} - 6 {(0)}^{2}$

Step 4.1.2.1.3

Raising $0$ to any positive power yields $0$ .

$0 + 4 \cdot 0 - 6 {(0)}^{2}$

Step 4.1.2.1.4

Multiply $4$ by $0$ .

$0 + 0 - 6 {(0)}^{2}$

Step 4.1.2.1.5

Raising $0$ to any positive power yields $0$ .

$0 + 0 - 6 \cdot 0$

Step 4.1.2.1.6

Multiply $- 6$ by $0$ .

$0 + 0 + 0$

Step 4.1.2.2

Simplify by adding numbers.

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Step 4.1.2.2.1

Add $0$ and $0$ .

$0 + 0$

Step 4.1.2.2.2

Add $0$ and $0$ .

$0$

Step 4.2

Evaluate at $x = - \frac{1 - \sqrt{5}}{2}$ .

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Step 4.2.1

Substitute $- \frac{1 - \sqrt{5}}{2}$ for $x$ .

$3 {(- \frac{1 - \sqrt{5}}{2})}^{4} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.2.1.1.1

Apply the product rule to $- \frac{1 - \sqrt{5}}{2}$ .

$3 ({(- 1)}^{4} {(\frac{1 - \sqrt{5}}{2})}^{4}) + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.1.2

Apply the product rule to $\frac{1 - \sqrt{5}}{2}$ .

$3 ({(- 1)}^{4} \frac{{(1 - \sqrt{5})}^{4}}{2^{4}}) + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.2

Raise $- 1$ to the power of $4$ .

$3 (1 \frac{{(1 - \sqrt{5})}^{4}}{2^{4}}) + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.3

Multiply $\frac{{(1 - \sqrt{5})}^{4}}{2^{4}}$ by $1$ .

$3 \frac{{(1 - \sqrt{5})}^{4}}{2^{4}} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.4

Raise $2$ to the power of $4$ .

$3 \frac{{(1 - \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.5

Use the Binomial Theorem.

$3 \frac{1^{4} + 4 \cdot 1^{3} (- \sqrt{5}) + 6 \cdot 1^{2} {(- \sqrt{5})}^{2} + 4 \cdot 1 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6

Simplify each term.

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Step 4.2.2.1.6.1

One to any power is one.

$3 \frac{1 + 4 \cdot 1^{3} (- \sqrt{5}) + 6 \cdot 1^{2} {(- \sqrt{5})}^{2} + 4 \cdot 1 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.2

One to any power is one.

$3 \frac{1 + 4 \cdot 1 (- \sqrt{5}) + 6 \cdot 1^{2} {(- \sqrt{5})}^{2} + 4 \cdot 1 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.3

Multiply $4$ by $1$ .

$3 \frac{1 + 4 (- \sqrt{5}) + 6 \cdot 1^{2} {(- \sqrt{5})}^{2} + 4 \cdot 1 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.4

Multiply $- 1$ by $4$ .

$3 \frac{1 - 4 \sqrt{5} + 6 \cdot 1^{2} {(- \sqrt{5})}^{2} + 4 \cdot 1 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.5

One to any power is one.

$3 \frac{1 - 4 \sqrt{5} + 6 \cdot 1 {(- \sqrt{5})}^{2} + 4 \cdot 1 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.6

Multiply $6$ by $1$ .

$3 \frac{1 - 4 \sqrt{5} + 6 {(- \sqrt{5})}^{2} + 4 \cdot 1 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.7

Apply the product rule to $- \sqrt{5}$ .

$3 \frac{1 - 4 \sqrt{5} + 6 ({(- 1)}^{2} {\sqrt{5}}^{2}) + 4 \cdot 1 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.8

Raise $- 1$ to the power of $2$ .

$3 \frac{1 - 4 \sqrt{5} + 6 (1 {\sqrt{5}}^{2}) + 4 \cdot 1 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.9

Multiply ${\sqrt{5}}^{2}$ by $1$ .

$3 \frac{1 - 4 \sqrt{5} + 6 {\sqrt{5}}^{2} + 4 \cdot 1 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.10

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.2.2.1.6.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$3 \frac{1 - 4 \sqrt{5} + 6 {(5^{\frac{1}{2}})}^{2} + 4 \cdot 1 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.10.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 \frac{1 - 4 \sqrt{5} + 6 \cdot 5^{\frac{1}{2} \cdot 2} + 4 \cdot 1 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.10.3

Combine $\frac{1}{2}$ and $2$ .

$3 \frac{1 - 4 \sqrt{5} + 6 \cdot 5^{\frac{2}{2}} + 4 \cdot 1 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.10.4

Cancel the common factor of $2$ .

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Step 4.2.2.1.6.10.4.1

Cancel the common factor.

$3 \frac{1 - 4 \sqrt{5} + 6 \cdot 5^{\frac{2}{2}} + 4 \cdot 1 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.10.4.2

Rewrite the expression.

$3 \frac{1 - 4 \sqrt{5} + 6 \cdot 5^{1} + 4 \cdot 1 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.10.5

Evaluate the exponent.

$3 \frac{1 - 4 \sqrt{5} + 6 \cdot 5 + 4 \cdot 1 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.11

Multiply $6$ by $5$ .

$3 \frac{1 - 4 \sqrt{5} + 30 + 4 \cdot 1 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.12

Multiply $4$ by $1$ .

$3 \frac{1 - 4 \sqrt{5} + 30 + 4 {(- \sqrt{5})}^{3} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.13

Apply the product rule to $- \sqrt{5}$ .

$3 \frac{1 - 4 \sqrt{5} + 30 + 4 ({(- 1)}^{3} {\sqrt{5}}^{3}) + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.14

Raise $- 1$ to the power of $3$ .

$3 \frac{1 - 4 \sqrt{5} + 30 + 4 (- {\sqrt{5}}^{3}) + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.15

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$3 \frac{1 - 4 \sqrt{5} + 30 + 4 (- \sqrt{5^{3}}) + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.16

Raise $5$ to the power of $3$ .

$3 \frac{1 - 4 \sqrt{5} + 30 + 4 (- \sqrt{125}) + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.17

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 4.2.2.1.6.17.1

Factor $25$ out of $125$ .

$3 \frac{1 - 4 \sqrt{5} + 30 + 4 (- \sqrt{25 (5)}) + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.17.2

Rewrite $25$ as $5^{2}$ .

$3 \frac{1 - 4 \sqrt{5} + 30 + 4 (- \sqrt{5^{2} \cdot 5}) + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.18

Pull terms out from under the radical.

$3 \frac{1 - 4 \sqrt{5} + 30 + 4 (- (5 \sqrt{5})) + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.19

Multiply $5$ by $- 1$ .

$3 \frac{1 - 4 \sqrt{5} + 30 + 4 (- 5 \sqrt{5}) + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.20

Multiply $- 5$ by $4$ .

$3 \frac{1 - 4 \sqrt{5} + 30 - 20 \sqrt{5} + {(- \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.21

Apply the product rule to $- \sqrt{5}$ .

$3 \frac{1 - 4 \sqrt{5} + 30 - 20 \sqrt{5} + {(- 1)}^{4} {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.22

Raise $- 1$ to the power of $4$ .

$3 \frac{1 - 4 \sqrt{5} + 30 - 20 \sqrt{5} + 1 {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.23

Multiply ${\sqrt{5}}^{4}$ by $1$ .

$3 \frac{1 - 4 \sqrt{5} + 30 - 20 \sqrt{5} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.24

Rewrite ${\sqrt{5}}^{4}$ as $5^{2}$ .

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Step 4.2.2.1.6.24.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$3 \frac{1 - 4 \sqrt{5} + 30 - 20 \sqrt{5} + {(5^{\frac{1}{2}})}^{4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.24.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 \frac{1 - 4 \sqrt{5} + 30 - 20 \sqrt{5} + 5^{\frac{1}{2} \cdot 4}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.24.3

Combine $\frac{1}{2}$ and $4$ .

$3 \frac{1 - 4 \sqrt{5} + 30 - 20 \sqrt{5} + 5^{\frac{4}{2}}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.24.4

Cancel the common factor of $4$ and $2$ .

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Step 4.2.2.1.6.24.4.1

Factor $2$ out of $4$ .

$3 \frac{1 - 4 \sqrt{5} + 30 - 20 \sqrt{5} + 5^{\frac{2 \cdot 2}{2}}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.24.4.2

Cancel the common factors.

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Step 4.2.2.1.6.24.4.2.1

Factor $2$ out of $2$ .

$3 \frac{1 - 4 \sqrt{5} + 30 - 20 \sqrt{5} + 5^{\frac{2 \cdot 2}{2 (1)}}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.24.4.2.2

Cancel the common factor.

$3 \frac{1 - 4 \sqrt{5} + 30 - 20 \sqrt{5} + 5^{\frac{2 \cdot 2}{2 \cdot 1}}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.24.4.2.3

Rewrite the expression.

$3 \frac{1 - 4 \sqrt{5} + 30 - 20 \sqrt{5} + 5^{\frac{2}{1}}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.24.4.2.4

Divide $2$ by $1$ .

$3 \frac{1 - 4 \sqrt{5} + 30 - 20 \sqrt{5} + 5^{2}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.6.25

Raise $5$ to the power of $2$ .

$3 \frac{1 - 4 \sqrt{5} + 30 - 20 \sqrt{5} + 25}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.7

Add $1$ and $30$ .

$3 \frac{31 - 4 \sqrt{5} - 20 \sqrt{5} + 25}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.8

Add $31$ and $25$ .

$3 \frac{56 - 4 \sqrt{5} - 20 \sqrt{5}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.9

Subtract $20 \sqrt{5}$ from $- 4 \sqrt{5}$ .

$3 \frac{56 - 24 \sqrt{5}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.10

Cancel the common factor of $56 - 24 \sqrt{5}$ and $16$ .

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Step 4.2.2.1.10.1

Factor $8$ out of $56$ .

$3 \frac{8 (7) - 24 \sqrt{5}}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.10.2

Factor $8$ out of $- 24 \sqrt{5}$ .

$3 \frac{8 (7) + 8 (- 3 \sqrt{5})}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.10.3

Factor $8$ out of $8 (7) + 8 (- 3 \sqrt{5})$ .

$3 \frac{8 (7 - 3 \sqrt{5})}{16} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.10.4

Cancel the common factors.

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Step 4.2.2.1.10.4.1

Factor $8$ out of $16$ .

$3 \frac{8 (7 - 3 \sqrt{5})}{8 \cdot 2} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.10.4.2

Cancel the common factor.

$3 \frac{8 (7 - 3 \sqrt{5})}{8 \cdot 2} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.10.4.3

Rewrite the expression.

$3 \frac{7 - 3 \sqrt{5}}{2} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.11

Combine $3$ and $\frac{7 - 3 \sqrt{5}}{2}$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + 4 {(- \frac{1 - \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.12

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.2.1.12.1

Apply the product rule to $- \frac{1 - \sqrt{5}}{2}$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + 4 ({(- 1)}^{3} {(\frac{1 - \sqrt{5}}{2})}^{3}) - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.12.2

Apply the product rule to $\frac{1 - \sqrt{5}}{2}$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + 4 ({(- 1)}^{3} \frac{{(1 - \sqrt{5})}^{3}}{2^{3}}) - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.13

Raise $- 1$ to the power of $3$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + 4 (- \frac{{(1 - \sqrt{5})}^{3}}{2^{3}}) - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.14

Raise $2$ to the power of $3$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + 4 (- \frac{{(1 - \sqrt{5})}^{3}}{8}) - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.15

Cancel the common factor of $4$ .

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Step 4.2.2.1.15.1

Move the leading negative in $- \frac{{(1 - \sqrt{5})}^{3}}{8}$ into the numerator.

$\frac{3 (7 - 3 \sqrt{5})}{2} + 4 \frac{- {(1 - \sqrt{5})}^{3}}{8} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.15.2

Factor $4$ out of $8$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + 4 \frac{- {(1 - \sqrt{5})}^{3}}{4 (2)} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.15.3

Cancel the common factor.

$\frac{3 (7 - 3 \sqrt{5})}{2} + 4 \frac{- {(1 - \sqrt{5})}^{3}}{4 \cdot 2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.15.4

Rewrite the expression.

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- {(1 - \sqrt{5})}^{3}}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.16

Use the Binomial Theorem.

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1^{3} + 3 \cdot 1^{2} (- \sqrt{5}) + 3 \cdot 1 {(- \sqrt{5})}^{2} + {(- \sqrt{5})}^{3})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17

Simplify each term.

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Step 4.2.2.1.17.1

One to any power is one.

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 + 3 \cdot 1^{2} (- \sqrt{5}) + 3 \cdot 1 {(- \sqrt{5})}^{2} + {(- \sqrt{5})}^{3})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.2

One to any power is one.

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 + 3 \cdot 1 (- \sqrt{5}) + 3 \cdot 1 {(- \sqrt{5})}^{2} + {(- \sqrt{5})}^{3})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.3

Multiply $3$ by $1$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 + 3 (- \sqrt{5}) + 3 \cdot 1 {(- \sqrt{5})}^{2} + {(- \sqrt{5})}^{3})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.4

Multiply $- 1$ by $3$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 3 \cdot 1 {(- \sqrt{5})}^{2} + {(- \sqrt{5})}^{3})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.5

Multiply $3$ by $1$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 3 {(- \sqrt{5})}^{2} + {(- \sqrt{5})}^{3})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.6

Apply the product rule to $- \sqrt{5}$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 3 ({(- 1)}^{2} {\sqrt{5}}^{2}) + {(- \sqrt{5})}^{3})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.7

Raise $- 1$ to the power of $2$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 3 (1 {\sqrt{5}}^{2}) + {(- \sqrt{5})}^{3})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.8

Multiply ${\sqrt{5}}^{2}$ by $1$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 3 {\sqrt{5}}^{2} + {(- \sqrt{5})}^{3})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.9

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.2.2.1.17.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 3 {(5^{\frac{1}{2}})}^{2} + {(- \sqrt{5})}^{3})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.9.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 3 \cdot 5^{\frac{1}{2} \cdot 2} + {(- \sqrt{5})}^{3})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.9.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 3 \cdot 5^{\frac{2}{2}} + {(- \sqrt{5})}^{3})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.9.4

Cancel the common factor of $2$ .

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Step 4.2.2.1.17.9.4.1

Cancel the common factor.

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 3 \cdot 5^{\frac{2}{2}} + {(- \sqrt{5})}^{3})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.9.4.2

Rewrite the expression.

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 3 \cdot 5^{1} + {(- \sqrt{5})}^{3})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.9.5

Evaluate the exponent.

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 3 \cdot 5 + {(- \sqrt{5})}^{3})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.10

Multiply $3$ by $5$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 15 + {(- \sqrt{5})}^{3})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.11

Apply the product rule to $- \sqrt{5}$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 15 + {(- 1)}^{3} {\sqrt{5}}^{3})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.12

Raise $- 1$ to the power of $3$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 15 - {\sqrt{5}}^{3})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.13

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 15 - \sqrt{5^{3}})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.14

Raise $5$ to the power of $3$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 15 - \sqrt{125})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.15

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 4.2.2.1.17.15.1

Factor $25$ out of $125$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 15 - \sqrt{25 (5)})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.15.2

Rewrite $25$ as $5^{2}$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 15 - \sqrt{5^{2} \cdot 5})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.16

Pull terms out from under the radical.

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 15 - (5 \sqrt{5}))}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.17.17

Multiply $5$ by $- 1$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (1 - 3 \sqrt{5} + 15 - 5 \sqrt{5})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.18

Add $1$ and $15$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (16 - 3 \sqrt{5} - 5 \sqrt{5})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.19

Subtract $5 \sqrt{5}$ from $- 3 \sqrt{5}$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (16 - 8 \sqrt{5})}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.20

Cancel the common factor of $16 - 8 \sqrt{5}$ and $2$ .

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Step 4.2.2.1.20.1

Factor $2$ out of $- (16 - 8 \sqrt{5})$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{2 (- (8 - 4 \sqrt{5}))}{2} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.20.2

Cancel the common factors.

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Step 4.2.2.1.20.2.1

Factor $2$ out of $2$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{2 (- (8 - 4 \sqrt{5}))}{2 (1)} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.20.2.2

Cancel the common factor.

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{2 (- (8 - 4 \sqrt{5}))}{2 \cdot 1} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.20.2.3

Rewrite the expression.

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- (8 - 4 \sqrt{5})}{1} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.20.2.4

Divide $- (8 - 4 \sqrt{5})$ by $1$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - (8 - 4 \sqrt{5}) - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.21

Apply the distributive property.

$\frac{3 (7 - 3 \sqrt{5})}{2} - 1 \cdot 8 - (- 4 \sqrt{5}) - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.22

Multiply $- 1$ by $8$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 - (- 4 \sqrt{5}) - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.23

Multiply $- 4$ by $- 1$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} - 6 {(- \frac{1 - \sqrt{5}}{2})}^{2}$

Step 4.2.2.1.24

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.2.1.24.1

Apply the product rule to $- \frac{1 - \sqrt{5}}{2}$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} - 6 ({(- 1)}^{2} {(\frac{1 - \sqrt{5}}{2})}^{2})$

Step 4.2.2.1.24.2

Apply the product rule to $\frac{1 - \sqrt{5}}{2}$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} - 6 ({(- 1)}^{2} \frac{{(1 - \sqrt{5})}^{2}}{2^{2}})$

Step 4.2.2.1.25

Raise $- 1$ to the power of $2$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} - 6 (1 \frac{{(1 - \sqrt{5})}^{2}}{2^{2}})$

Step 4.2.2.1.26

Multiply $\frac{{(1 - \sqrt{5})}^{2}}{2^{2}}$ by $1$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} - 6 \frac{{(1 - \sqrt{5})}^{2}}{2^{2}}$

Step 4.2.2.1.27

Raise $2$ to the power of $2$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} - 6 \frac{{(1 - \sqrt{5})}^{2}}{4}$

Step 4.2.2.1.28

Cancel the common factor of $2$ .

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Step 4.2.2.1.28.1

Factor $2$ out of $- 6$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + 2 (- 3) \frac{{(1 - \sqrt{5})}^{2}}{4}$

Step 4.2.2.1.28.2

Factor $2$ out of $4$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + 2 \cdot - 3 \frac{{(1 - \sqrt{5})}^{2}}{2 \cdot 2}$

Step 4.2.2.1.28.3

Cancel the common factor.

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + 2 \cdot - 3 \frac{{(1 - \sqrt{5})}^{2}}{2 \cdot 2}$

Step 4.2.2.1.28.4

Rewrite the expression.

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} - 3 \frac{{(1 - \sqrt{5})}^{2}}{2}$

Step 4.2.2.1.29

Combine $- 3$ and $\frac{{(1 - \sqrt{5})}^{2}}{2}$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 {(1 - \sqrt{5})}^{2}}{2}$

Step 4.2.2.1.30

Rewrite ${(1 - \sqrt{5})}^{2}$ as $(1 - \sqrt{5}) (1 - \sqrt{5})$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 ((1 - \sqrt{5}) (1 - \sqrt{5}))}{2}$

Step 4.2.2.1.31

Expand $(1 - \sqrt{5}) (1 - \sqrt{5})$ using the FOIL Method.

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Step 4.2.2.1.31.1

Apply the distributive property.

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (1 (1 - \sqrt{5}) - \sqrt{5} (1 - \sqrt{5}))}{2}$

Step 4.2.2.1.31.2

Apply the distributive property.

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (1 \cdot 1 + 1 (- \sqrt{5}) - \sqrt{5} (1 - \sqrt{5}))}{2}$

Step 4.2.2.1.31.3

Apply the distributive property.

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (1 \cdot 1 + 1 (- \sqrt{5}) - \sqrt{5} \cdot 1 - \sqrt{5} (- \sqrt{5}))}{2}$

Step 4.2.2.1.32

Simplify and combine like terms.

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Step 4.2.2.1.32.1

Simplify each term.

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Step 4.2.2.1.32.1.1

Multiply $1$ by $1$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (1 + 1 (- \sqrt{5}) - \sqrt{5} \cdot 1 - \sqrt{5} (- \sqrt{5}))}{2}$

Step 4.2.2.1.32.1.2

Multiply $- \sqrt{5}$ by $1$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (1 - \sqrt{5} - \sqrt{5} \cdot 1 - \sqrt{5} (- \sqrt{5}))}{2}$

Step 4.2.2.1.32.1.3

Multiply $- 1$ by $1$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (1 - \sqrt{5} - \sqrt{5} - \sqrt{5} (- \sqrt{5}))}{2}$

Step 4.2.2.1.32.1.4

Multiply $- \sqrt{5} (- \sqrt{5})$ .

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Step 4.2.2.1.32.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (1 - \sqrt{5} - \sqrt{5} + 1 \sqrt{5} \sqrt{5})}{2}$

Step 4.2.2.1.32.1.4.2

Multiply $\sqrt{5}$ by $1$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (1 - \sqrt{5} - \sqrt{5} + \sqrt{5} \sqrt{5})}{2}$

Step 4.2.2.1.32.1.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (1 - \sqrt{5} - \sqrt{5} + {\sqrt{5}}^{1} \sqrt{5})}{2}$

Step 4.2.2.1.32.1.4.4

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (1 - \sqrt{5} - \sqrt{5} + {\sqrt{5}}^{1} {\sqrt{5}}^{1})}{2}$

Step 4.2.2.1.32.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (1 - \sqrt{5} - \sqrt{5} + {\sqrt{5}}^{1 + 1})}{2}$

Step 4.2.2.1.32.1.4.6

Add $1$ and $1$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (1 - \sqrt{5} - \sqrt{5} + {\sqrt{5}}^{2})}{2}$

Step 4.2.2.1.32.1.5

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.2.2.1.32.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (1 - \sqrt{5} - \sqrt{5} + {(5^{\frac{1}{2}})}^{2})}{2}$

Step 4.2.2.1.32.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (1 - \sqrt{5} - \sqrt{5} + 5^{\frac{1}{2} \cdot 2})}{2}$

Step 4.2.2.1.32.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (1 - \sqrt{5} - \sqrt{5} + 5^{\frac{2}{2}})}{2}$

Step 4.2.2.1.32.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.2.1.32.1.5.4.1

Cancel the common factor.

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (1 - \sqrt{5} - \sqrt{5} + 5^{\frac{2}{2}})}{2}$

Step 4.2.2.1.32.1.5.4.2

Rewrite the expression.

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (1 - \sqrt{5} - \sqrt{5} + 5^{1})}{2}$

Step 4.2.2.1.32.1.5.5

Evaluate the exponent.

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (1 - \sqrt{5} - \sqrt{5} + 5)}{2}$

Step 4.2.2.1.32.2

Add $1$ and $5$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (6 - \sqrt{5} - \sqrt{5})}{2}$

Step 4.2.2.1.32.3

Subtract $\sqrt{5}$ from $- \sqrt{5}$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (6 - 2 \sqrt{5})}{2}$

Step 4.2.2.1.33

Cancel the common factor of $6 - 2 \sqrt{5}$ and $2$ .

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Step 4.2.2.1.33.1

Factor $2$ out of $- 3 (6 - 2 \sqrt{5})$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{2 (- 3 (3 - \sqrt{5}))}{2}$

Step 4.2.2.1.33.2

Cancel the common factors.

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Step 4.2.2.1.33.2.1

Factor $2$ out of $2$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{2 (- 3 (3 - \sqrt{5}))}{2 (1)}$

Step 4.2.2.1.33.2.2

Cancel the common factor.

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{2 (- 3 (3 - \sqrt{5}))}{2 \cdot 1}$

Step 4.2.2.1.33.2.3

Rewrite the expression.

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} + \frac{- 3 (3 - \sqrt{5})}{1}$

Step 4.2.2.1.33.2.4

Divide $- 3 (3 - \sqrt{5})$ by $1$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} - 3 (3 - \sqrt{5})$

Step 4.2.2.1.34

Apply the distributive property.

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} - 3 \cdot 3 - 3 (- \sqrt{5})$

Step 4.2.2.1.35

Multiply $- 3$ by $3$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} - 9 - 3 (- \sqrt{5})$

Step 4.2.2.1.36

Multiply $- 1$ by $- 3$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 + 4 \sqrt{5} - 9 + 3 \sqrt{5}$

Step 4.2.2.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} - 8 \cdot \frac{2}{2} + 4 \sqrt{5} - 9 + 3 \sqrt{5}$

Step 4.2.2.3

Combine fractions.

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Step 4.2.2.3.1

Combine $- 8$ and $\frac{2}{2}$ .

$\frac{3 (7 - 3 \sqrt{5})}{2} + \frac{- 8 \cdot 2}{2} + 4 \sqrt{5} - 9 + 3 \sqrt{5}$

Step 4.2.2.3.2

Simplify the expression.

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Step 4.2.2.3.2.1

Combine the numerators over the common denominator.

$\frac{3 (7 - 3 \sqrt{5}) - 8 \cdot 2}{2} + 4 \sqrt{5} - 9 + 3 \sqrt{5}$

Step 4.2.2.3.2.2

Multiply $- 8$ by $2$ .

$\frac{3 (7 - 3 \sqrt{5}) - 16}{2} + 4 \sqrt{5} - 9 + 3 \sqrt{5}$

Step 4.2.2.4

Simplify the numerator.

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Step 4.2.2.4.1

Apply the distributive property.

$\frac{3 \cdot 7 + 3 (- 3 \sqrt{5}) - 16}{2} + 4 \sqrt{5} - 9 + 3 \sqrt{5}$

Step 4.2.2.4.2

Multiply $3$ by $7$ .

$\frac{21 + 3 (- 3 \sqrt{5}) - 16}{2} + 4 \sqrt{5} - 9 + 3 \sqrt{5}$

Step 4.2.2.4.3

Multiply $- 3$ by $3$ .

$\frac{21 - 9 \sqrt{5} - 16}{2} + 4 \sqrt{5} - 9 + 3 \sqrt{5}$

Step 4.2.2.4.4

Subtract $16$ from $21$ .

$\frac{5 - 9 \sqrt{5}}{2} + 4 \sqrt{5} - 9 + 3 \sqrt{5}$

Step 4.2.2.5

To write $4 \sqrt{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5 - 9 \sqrt{5}}{2} + 4 \sqrt{5} \cdot \frac{2}{2} - 9 + 3 \sqrt{5}$

Step 4.2.2.6

Combine fractions.

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Step 4.2.2.6.1

Combine $4 \sqrt{5}$ and $\frac{2}{2}$ .

$\frac{5 - 9 \sqrt{5}}{2} + \frac{4 \sqrt{5} \cdot 2}{2} - 9 + 3 \sqrt{5}$

Step 4.2.2.6.2

Combine the numerators over the common denominator.

$\frac{5 - 9 \sqrt{5} + 4 \sqrt{5} \cdot 2}{2} - 9 + 3 \sqrt{5}$

Step 4.2.2.7

Simplify the numerator.

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Step 4.2.2.7.1

Multiply $2$ by $4$ .

$\frac{5 - 9 \sqrt{5} + 8 \sqrt{5}}{2} - 9 + 3 \sqrt{5}$

Step 4.2.2.7.2

Add $- 9 \sqrt{5}$ and $8 \sqrt{5}$ .

$\frac{5 - \sqrt{5}}{2} - 9 + 3 \sqrt{5}$

Step 4.2.2.8

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5 - \sqrt{5}}{2} - 9 \cdot \frac{2}{2} + 3 \sqrt{5}$

Step 4.2.2.9

Combine $- 9$ and $\frac{2}{2}$ .

$\frac{5 - \sqrt{5}}{2} + \frac{- 9 \cdot 2}{2} + 3 \sqrt{5}$

Step 4.2.2.10

Simplify the expression.

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Step 4.2.2.10.1

Combine the numerators over the common denominator.

$\frac{5 - \sqrt{5} - 9 \cdot 2}{2} + 3 \sqrt{5}$

Step 4.2.2.10.2

Multiply $- 9$ by $2$ .

$\frac{5 - \sqrt{5} - 18}{2} + 3 \sqrt{5}$

Step 4.2.2.10.3

Subtract $18$ from $5$ .

$\frac{- 13 - \sqrt{5}}{2} + 3 \sqrt{5}$

Step 4.2.2.11

To write $3 \sqrt{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- 13 - \sqrt{5}}{2} + 3 \sqrt{5} \cdot \frac{2}{2}$

Step 4.2.2.12

Combine fractions.

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Step 4.2.2.12.1

Combine $3 \sqrt{5}$ and $\frac{2}{2}$ .

$\frac{- 13 - \sqrt{5}}{2} + \frac{3 \sqrt{5} \cdot 2}{2}$

Step 4.2.2.12.2

Combine the numerators over the common denominator.

$\frac{- 13 - \sqrt{5} + 3 \sqrt{5} \cdot 2}{2}$

Step 4.2.2.13

Simplify the numerator.

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Step 4.2.2.13.1

Multiply $2$ by $3$ .

$\frac{- 13 - \sqrt{5} + 6 \sqrt{5}}{2}$

Step 4.2.2.13.2

Add $- \sqrt{5}$ and $6 \sqrt{5}$ .

$\frac{- 13 + 5 \sqrt{5}}{2}$

Step 4.2.2.14

Simplify with factoring out.

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Step 4.2.2.14.1

Rewrite $- 13$ as $- 1 (13)$ .

$\frac{- 1 (13) + 5 \sqrt{5}}{2}$

Step 4.2.2.14.2

Factor $- 1$ out of $5 \sqrt{5}$ .

$\frac{- 1 (13) - (- 5 \sqrt{5})}{2}$

Step 4.2.2.14.3

Factor $- 1$ out of $- 1 (13) - (- 5 \sqrt{5})$ .

$\frac{- 1 (13 - 5 \sqrt{5})}{2}$

Step 4.2.2.14.4

Move the negative in front of the fraction.

$- \frac{13 - 5 \sqrt{5}}{2}$

Step 4.3

Evaluate at $x = - \frac{1 + \sqrt{5}}{2}$ .

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Step 4.3.1

Substitute $- \frac{1 + \sqrt{5}}{2}$ for $x$ .

$3 {(- \frac{1 + \sqrt{5}}{2})}^{4} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.3.2.1.1.1

Apply the product rule to $- \frac{1 + \sqrt{5}}{2}$ .

$3 ({(- 1)}^{4} {(\frac{1 + \sqrt{5}}{2})}^{4}) + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.1.2

Apply the product rule to $\frac{1 + \sqrt{5}}{2}$ .

$3 ({(- 1)}^{4} \frac{{(1 + \sqrt{5})}^{4}}{2^{4}}) + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.2

Raise $- 1$ to the power of $4$ .

$3 (1 \frac{{(1 + \sqrt{5})}^{4}}{2^{4}}) + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.3

Multiply $\frac{{(1 + \sqrt{5})}^{4}}{2^{4}}$ by $1$ .

$3 \frac{{(1 + \sqrt{5})}^{4}}{2^{4}} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.4

Raise $2$ to the power of $4$ .

$3 \frac{{(1 + \sqrt{5})}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.5

Use the Binomial Theorem.

$3 \frac{1^{4} + 4 \cdot 1^{3} \sqrt{5} + 6 \cdot 1^{2} {\sqrt{5}}^{2} + 4 \cdot 1 {\sqrt{5}}^{3} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6

Simplify each term.

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Step 4.3.2.1.6.1

One to any power is one.

$3 \frac{1 + 4 \cdot 1^{3} \sqrt{5} + 6 \cdot 1^{2} {\sqrt{5}}^{2} + 4 \cdot 1 {\sqrt{5}}^{3} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.2

One to any power is one.

$3 \frac{1 + 4 \cdot 1 \sqrt{5} + 6 \cdot 1^{2} {\sqrt{5}}^{2} + 4 \cdot 1 {\sqrt{5}}^{3} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.3

Multiply $4$ by $1$ .

$3 \frac{1 + 4 \sqrt{5} + 6 \cdot 1^{2} {\sqrt{5}}^{2} + 4 \cdot 1 {\sqrt{5}}^{3} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.4

One to any power is one.

$3 \frac{1 + 4 \sqrt{5} + 6 \cdot 1 {\sqrt{5}}^{2} + 4 \cdot 1 {\sqrt{5}}^{3} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.5

Multiply $6$ by $1$ .

$3 \frac{1 + 4 \sqrt{5} + 6 {\sqrt{5}}^{2} + 4 \cdot 1 {\sqrt{5}}^{3} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.3.2.1.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$3 \frac{1 + 4 \sqrt{5} + 6 {(5^{\frac{1}{2}})}^{2} + 4 \cdot 1 {\sqrt{5}}^{3} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 \frac{1 + 4 \sqrt{5} + 6 \cdot 5^{\frac{1}{2} \cdot 2} + 4 \cdot 1 {\sqrt{5}}^{3} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.6.3

Combine $\frac{1}{2}$ and $2$ .

$3 \frac{1 + 4 \sqrt{5} + 6 \cdot 5^{\frac{2}{2}} + 4 \cdot 1 {\sqrt{5}}^{3} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.6.4

Cancel the common factor of $2$ .

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Step 4.3.2.1.6.6.4.1

Cancel the common factor.

$3 \frac{1 + 4 \sqrt{5} + 6 \cdot 5^{\frac{2}{2}} + 4 \cdot 1 {\sqrt{5}}^{3} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.6.4.2

Rewrite the expression.

$3 \frac{1 + 4 \sqrt{5} + 6 \cdot 5^{1} + 4 \cdot 1 {\sqrt{5}}^{3} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.6.5

Evaluate the exponent.

$3 \frac{1 + 4 \sqrt{5} + 6 \cdot 5 + 4 \cdot 1 {\sqrt{5}}^{3} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.7

Multiply $6$ by $5$ .

$3 \frac{1 + 4 \sqrt{5} + 30 + 4 \cdot 1 {\sqrt{5}}^{3} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.8

Multiply $4$ by $1$ .

$3 \frac{1 + 4 \sqrt{5} + 30 + 4 {\sqrt{5}}^{3} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.9

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$3 \frac{1 + 4 \sqrt{5} + 30 + 4 \sqrt{5^{3}} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.10

Raise $5$ to the power of $3$ .

$3 \frac{1 + 4 \sqrt{5} + 30 + 4 \sqrt{125} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.11

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 4.3.2.1.6.11.1

Factor $25$ out of $125$ .

$3 \frac{1 + 4 \sqrt{5} + 30 + 4 \sqrt{25 (5)} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.11.2

Rewrite $25$ as $5^{2}$ .

$3 \frac{1 + 4 \sqrt{5} + 30 + 4 \sqrt{5^{2} \cdot 5} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.12

Pull terms out from under the radical.

$3 \frac{1 + 4 \sqrt{5} + 30 + 4 (5 \sqrt{5}) + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.13

Multiply $5$ by $4$ .

$3 \frac{1 + 4 \sqrt{5} + 30 + 20 \sqrt{5} + {\sqrt{5}}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.14

Rewrite ${\sqrt{5}}^{4}$ as $5^{2}$ .

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Step 4.3.2.1.6.14.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$3 \frac{1 + 4 \sqrt{5} + 30 + 20 \sqrt{5} + {(5^{\frac{1}{2}})}^{4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.14.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 \frac{1 + 4 \sqrt{5} + 30 + 20 \sqrt{5} + 5^{\frac{1}{2} \cdot 4}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.14.3

Combine $\frac{1}{2}$ and $4$ .

$3 \frac{1 + 4 \sqrt{5} + 30 + 20 \sqrt{5} + 5^{\frac{4}{2}}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.14.4

Cancel the common factor of $4$ and $2$ .

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Step 4.3.2.1.6.14.4.1

Factor $2$ out of $4$ .

$3 \frac{1 + 4 \sqrt{5} + 30 + 20 \sqrt{5} + 5^{\frac{2 \cdot 2}{2}}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.14.4.2

Cancel the common factors.

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Step 4.3.2.1.6.14.4.2.1

Factor $2$ out of $2$ .

$3 \frac{1 + 4 \sqrt{5} + 30 + 20 \sqrt{5} + 5^{\frac{2 \cdot 2}{2 (1)}}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.14.4.2.2

Cancel the common factor.

$3 \frac{1 + 4 \sqrt{5} + 30 + 20 \sqrt{5} + 5^{\frac{2 \cdot 2}{2 \cdot 1}}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.14.4.2.3

Rewrite the expression.

$3 \frac{1 + 4 \sqrt{5} + 30 + 20 \sqrt{5} + 5^{\frac{2}{1}}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.14.4.2.4

Divide $2$ by $1$ .

$3 \frac{1 + 4 \sqrt{5} + 30 + 20 \sqrt{5} + 5^{2}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.6.15

Raise $5$ to the power of $2$ .

$3 \frac{1 + 4 \sqrt{5} + 30 + 20 \sqrt{5} + 25}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.7

Add $1$ and $30$ .

$3 \frac{31 + 4 \sqrt{5} + 20 \sqrt{5} + 25}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.8

Add $31$ and $25$ .

$3 \frac{56 + 4 \sqrt{5} + 20 \sqrt{5}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.9

Add $4 \sqrt{5}$ and $20 \sqrt{5}$ .

$3 \frac{56 + 24 \sqrt{5}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.10

Cancel the common factor of $56 + 24 \sqrt{5}$ and $16$ .

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Step 4.3.2.1.10.1

Factor $8$ out of $56$ .

$3 \frac{8 (7) + 24 \sqrt{5}}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.10.2

Factor $8$ out of $24 \sqrt{5}$ .

$3 \frac{8 (7) + 8 (3 \sqrt{5})}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.10.3

Factor $8$ out of $8 (7) + 8 (3 \sqrt{5})$ .

$3 \frac{8 (7 + 3 \sqrt{5})}{16} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.10.4

Cancel the common factors.

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Step 4.3.2.1.10.4.1

Factor $8$ out of $16$ .

$3 \frac{8 (7 + 3 \sqrt{5})}{8 \cdot 2} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.10.4.2

Cancel the common factor.

$3 \frac{8 (7 + 3 \sqrt{5})}{8 \cdot 2} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.10.4.3

Rewrite the expression.

$3 \frac{7 + 3 \sqrt{5}}{2} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.11

Combine $3$ and $\frac{7 + 3 \sqrt{5}}{2}$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + 4 {(- \frac{1 + \sqrt{5}}{2})}^{3} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.12

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.3.2.1.12.1

Apply the product rule to $- \frac{1 + \sqrt{5}}{2}$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + 4 ({(- 1)}^{3} {(\frac{1 + \sqrt{5}}{2})}^{3}) - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.12.2

Apply the product rule to $\frac{1 + \sqrt{5}}{2}$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + 4 ({(- 1)}^{3} \frac{{(1 + \sqrt{5})}^{3}}{2^{3}}) - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.13

Raise $- 1$ to the power of $3$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + 4 (- \frac{{(1 + \sqrt{5})}^{3}}{2^{3}}) - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.14

Raise $2$ to the power of $3$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + 4 (- \frac{{(1 + \sqrt{5})}^{3}}{8}) - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.15

Cancel the common factor of $4$ .

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Step 4.3.2.1.15.1

Move the leading negative in $- \frac{{(1 + \sqrt{5})}^{3}}{8}$ into the numerator.

$\frac{3 (7 + 3 \sqrt{5})}{2} + 4 \frac{- {(1 + \sqrt{5})}^{3}}{8} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.15.2

Factor $4$ out of $8$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + 4 \frac{- {(1 + \sqrt{5})}^{3}}{4 (2)} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.15.3

Cancel the common factor.

$\frac{3 (7 + 3 \sqrt{5})}{2} + 4 \frac{- {(1 + \sqrt{5})}^{3}}{4 \cdot 2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.15.4

Rewrite the expression.

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- {(1 + \sqrt{5})}^{3}}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.16

Use the Binomial Theorem.

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (1^{3} + 3 \cdot 1^{2} \sqrt{5} + 3 \cdot 1 {\sqrt{5}}^{2} + {\sqrt{5}}^{3})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.17

Simplify each term.

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Step 4.3.2.1.17.1

One to any power is one.

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (1 + 3 \cdot 1^{2} \sqrt{5} + 3 \cdot 1 {\sqrt{5}}^{2} + {\sqrt{5}}^{3})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.17.2

One to any power is one.

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (1 + 3 \cdot 1 \sqrt{5} + 3 \cdot 1 {\sqrt{5}}^{2} + {\sqrt{5}}^{3})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.17.3

Multiply $3$ by $1$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (1 + 3 \sqrt{5} + 3 \cdot 1 {\sqrt{5}}^{2} + {\sqrt{5}}^{3})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.17.4

Multiply $3$ by $1$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (1 + 3 \sqrt{5} + 3 {\sqrt{5}}^{2} + {\sqrt{5}}^{3})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.17.5

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.3.2.1.17.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (1 + 3 \sqrt{5} + 3 {(5^{\frac{1}{2}})}^{2} + {\sqrt{5}}^{3})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.17.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (1 + 3 \sqrt{5} + 3 \cdot 5^{\frac{1}{2} \cdot 2} + {\sqrt{5}}^{3})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.17.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (1 + 3 \sqrt{5} + 3 \cdot 5^{\frac{2}{2}} + {\sqrt{5}}^{3})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.17.5.4

Cancel the common factor of $2$ .

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Step 4.3.2.1.17.5.4.1

Cancel the common factor.

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (1 + 3 \sqrt{5} + 3 \cdot 5^{\frac{2}{2}} + {\sqrt{5}}^{3})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.17.5.4.2

Rewrite the expression.

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (1 + 3 \sqrt{5} + 3 \cdot 5^{1} + {\sqrt{5}}^{3})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.17.5.5

Evaluate the exponent.

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (1 + 3 \sqrt{5} + 3 \cdot 5 + {\sqrt{5}}^{3})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.17.6

Multiply $3$ by $5$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (1 + 3 \sqrt{5} + 15 + {\sqrt{5}}^{3})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.17.7

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (1 + 3 \sqrt{5} + 15 + \sqrt{5^{3}})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.17.8

Raise $5$ to the power of $3$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (1 + 3 \sqrt{5} + 15 + \sqrt{125})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.17.9

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 4.3.2.1.17.9.1

Factor $25$ out of $125$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (1 + 3 \sqrt{5} + 15 + \sqrt{25 (5)})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.17.9.2

Rewrite $25$ as $5^{2}$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (1 + 3 \sqrt{5} + 15 + \sqrt{5^{2} \cdot 5})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.17.10

Pull terms out from under the radical.

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (1 + 3 \sqrt{5} + 15 + 5 \sqrt{5})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.18

Add $1$ and $15$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (16 + 3 \sqrt{5} + 5 \sqrt{5})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.19

Add $3 \sqrt{5}$ and $5 \sqrt{5}$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (16 + 8 \sqrt{5})}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.20

Cancel the common factor of $16 + 8 \sqrt{5}$ and $2$ .

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Step 4.3.2.1.20.1

Factor $2$ out of $- (16 + 8 \sqrt{5})$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{2 (- (8 + 4 \sqrt{5}))}{2} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.20.2

Cancel the common factors.

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Step 4.3.2.1.20.2.1

Factor $2$ out of $2$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{2 (- (8 + 4 \sqrt{5}))}{2 (1)} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.20.2.2

Cancel the common factor.

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{2 (- (8 + 4 \sqrt{5}))}{2 \cdot 1} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.20.2.3

Rewrite the expression.

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- (8 + 4 \sqrt{5})}{1} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.20.2.4

Divide $- (8 + 4 \sqrt{5})$ by $1$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - (8 + 4 \sqrt{5}) - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.21

Apply the distributive property.

$\frac{3 (7 + 3 \sqrt{5})}{2} - 1 \cdot 8 - (4 \sqrt{5}) - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.22

Multiply $- 1$ by $8$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - (4 \sqrt{5}) - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.23

Multiply $4$ by $- 1$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} - 6 {(- \frac{1 + \sqrt{5}}{2})}^{2}$

Step 4.3.2.1.24

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.3.2.1.24.1

Apply the product rule to $- \frac{1 + \sqrt{5}}{2}$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} - 6 ({(- 1)}^{2} {(\frac{1 + \sqrt{5}}{2})}^{2})$

Step 4.3.2.1.24.2

Apply the product rule to $\frac{1 + \sqrt{5}}{2}$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} - 6 ({(- 1)}^{2} \frac{{(1 + \sqrt{5})}^{2}}{2^{2}})$

Step 4.3.2.1.25

Raise $- 1$ to the power of $2$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} - 6 (1 \frac{{(1 + \sqrt{5})}^{2}}{2^{2}})$

Step 4.3.2.1.26

Multiply $\frac{{(1 + \sqrt{5})}^{2}}{2^{2}}$ by $1$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} - 6 \frac{{(1 + \sqrt{5})}^{2}}{2^{2}}$

Step 4.3.2.1.27

Raise $2$ to the power of $2$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} - 6 \frac{{(1 + \sqrt{5})}^{2}}{4}$

Step 4.3.2.1.28

Cancel the common factor of $2$ .

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Step 4.3.2.1.28.1

Factor $2$ out of $- 6$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + 2 (- 3) \frac{{(1 + \sqrt{5})}^{2}}{4}$

Step 4.3.2.1.28.2

Factor $2$ out of $4$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + 2 \cdot - 3 \frac{{(1 + \sqrt{5})}^{2}}{2 \cdot 2}$

Step 4.3.2.1.28.3

Cancel the common factor.

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + 2 \cdot - 3 \frac{{(1 + \sqrt{5})}^{2}}{2 \cdot 2}$

Step 4.3.2.1.28.4

Rewrite the expression.

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} - 3 \frac{{(1 + \sqrt{5})}^{2}}{2}$

Step 4.3.2.1.29

Combine $- 3$ and $\frac{{(1 + \sqrt{5})}^{2}}{2}$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + \frac{- 3 {(1 + \sqrt{5})}^{2}}{2}$

Step 4.3.2.1.30

Rewrite ${(1 + \sqrt{5})}^{2}$ as $(1 + \sqrt{5}) (1 + \sqrt{5})$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + \frac{- 3 ((1 + \sqrt{5}) (1 + \sqrt{5}))}{2}$

Step 4.3.2.1.31

Expand $(1 + \sqrt{5}) (1 + \sqrt{5})$ using the FOIL Method.

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Step 4.3.2.1.31.1

Apply the distributive property.

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + \frac{- 3 (1 (1 + \sqrt{5}) + \sqrt{5} (1 + \sqrt{5}))}{2}$

Step 4.3.2.1.31.2

Apply the distributive property.

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + \frac{- 3 (1 \cdot 1 + 1 \sqrt{5} + \sqrt{5} (1 + \sqrt{5}))}{2}$

Step 4.3.2.1.31.3

Apply the distributive property.

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + \frac{- 3 (1 \cdot 1 + 1 \sqrt{5} + \sqrt{5} \cdot 1 + \sqrt{5} \sqrt{5})}{2}$

Step 4.3.2.1.32

Simplify and combine like terms.

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Step 4.3.2.1.32.1

Simplify each term.

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Step 4.3.2.1.32.1.1

Multiply $1$ by $1$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + \frac{- 3 (1 + 1 \sqrt{5} + \sqrt{5} \cdot 1 + \sqrt{5} \sqrt{5})}{2}$

Step 4.3.2.1.32.1.2

Multiply $\sqrt{5}$ by $1$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + \frac{- 3 (1 + \sqrt{5} + \sqrt{5} \cdot 1 + \sqrt{5} \sqrt{5})}{2}$

Step 4.3.2.1.32.1.3

Multiply $\sqrt{5}$ by $1$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + \frac{- 3 (1 + \sqrt{5} + \sqrt{5} + \sqrt{5} \sqrt{5})}{2}$

Step 4.3.2.1.32.1.4

Combine using the product rule for radicals.

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + \frac{- 3 (1 + \sqrt{5} + \sqrt{5} + \sqrt{5 \cdot 5})}{2}$

Step 4.3.2.1.32.1.5

Multiply $5$ by $5$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + \frac{- 3 (1 + \sqrt{5} + \sqrt{5} + \sqrt{25})}{2}$

Step 4.3.2.1.32.1.6

Rewrite $25$ as $5^{2}$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + \frac{- 3 (1 + \sqrt{5} + \sqrt{5} + \sqrt{5^{2}})}{2}$

Step 4.3.2.1.32.1.7

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + \frac{- 3 (1 + \sqrt{5} + \sqrt{5} + 5)}{2}$

Step 4.3.2.1.32.2

Add $1$ and $5$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + \frac{- 3 (6 + \sqrt{5} + \sqrt{5})}{2}$

Step 4.3.2.1.32.3

Add $\sqrt{5}$ and $\sqrt{5}$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + \frac{- 3 (6 + 2 \sqrt{5})}{2}$

Step 4.3.2.1.33

Cancel the common factor of $6 + 2 \sqrt{5}$ and $2$ .

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Step 4.3.2.1.33.1

Factor $2$ out of $- 3 (6 + 2 \sqrt{5})$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + \frac{2 (- 3 (3 + \sqrt{5}))}{2}$

Step 4.3.2.1.33.2

Cancel the common factors.

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Step 4.3.2.1.33.2.1

Factor $2$ out of $2$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + \frac{2 (- 3 (3 + \sqrt{5}))}{2 (1)}$

Step 4.3.2.1.33.2.2

Cancel the common factor.

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + \frac{2 (- 3 (3 + \sqrt{5}))}{2 \cdot 1}$

Step 4.3.2.1.33.2.3

Rewrite the expression.

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} + \frac{- 3 (3 + \sqrt{5})}{1}$

Step 4.3.2.1.33.2.4

Divide $- 3 (3 + \sqrt{5})$ by $1$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} - 3 (3 + \sqrt{5})$

Step 4.3.2.1.34

Apply the distributive property.

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} - 3 \cdot 3 - 3 \sqrt{5}$

Step 4.3.2.1.35

Multiply $- 3$ by $3$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 - 4 \sqrt{5} - 9 - 3 \sqrt{5}$

Step 4.3.2.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} - 8 \cdot \frac{2}{2} - 4 \sqrt{5} - 9 - 3 \sqrt{5}$

Step 4.3.2.3

Combine fractions.

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Step 4.3.2.3.1

Combine $- 8$ and $\frac{2}{2}$ .

$\frac{3 (7 + 3 \sqrt{5})}{2} + \frac{- 8 \cdot 2}{2} - 4 \sqrt{5} - 9 - 3 \sqrt{5}$

Step 4.3.2.3.2

Simplify the expression.

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Step 4.3.2.3.2.1

Combine the numerators over the common denominator.

$\frac{3 (7 + 3 \sqrt{5}) - 8 \cdot 2}{2} - 4 \sqrt{5} - 9 - 3 \sqrt{5}$

Step 4.3.2.3.2.2

Multiply $- 8$ by $2$ .

$\frac{3 (7 + 3 \sqrt{5}) - 16}{2} - 4 \sqrt{5} - 9 - 3 \sqrt{5}$

Step 4.3.2.4

Simplify the numerator.

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Step 4.3.2.4.1

Apply the distributive property.

$\frac{3 \cdot 7 + 3 (3 \sqrt{5}) - 16}{2} - 4 \sqrt{5} - 9 - 3 \sqrt{5}$

Step 4.3.2.4.2

Multiply $3$ by $7$ .

$\frac{21 + 3 (3 \sqrt{5}) - 16}{2} - 4 \sqrt{5} - 9 - 3 \sqrt{5}$

Step 4.3.2.4.3

Multiply $3$ by $3$ .

$\frac{21 + 9 \sqrt{5} - 16}{2} - 4 \sqrt{5} - 9 - 3 \sqrt{5}$

Step 4.3.2.4.4

Subtract $16$ from $21$ .

$\frac{5 + 9 \sqrt{5}}{2} - 4 \sqrt{5} - 9 - 3 \sqrt{5}$

Step 4.3.2.5

To write $- 4 \sqrt{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5 + 9 \sqrt{5}}{2} - 4 \sqrt{5} \cdot \frac{2}{2} - 9 - 3 \sqrt{5}$

Step 4.3.2.6

Combine fractions.

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Step 4.3.2.6.1

Combine $- 4 \sqrt{5}$ and $\frac{2}{2}$ .

$\frac{5 + 9 \sqrt{5}}{2} + \frac{- 4 \sqrt{5} \cdot 2}{2} - 9 - 3 \sqrt{5}$

Step 4.3.2.6.2

Combine the numerators over the common denominator.

$\frac{5 + 9 \sqrt{5} - 4 \sqrt{5} \cdot 2}{2} - 9 - 3 \sqrt{5}$

Step 4.3.2.7

Simplify the numerator.

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Step 4.3.2.7.1

Multiply $2$ by $- 4$ .

$\frac{5 + 9 \sqrt{5} - 8 \sqrt{5}}{2} - 9 - 3 \sqrt{5}$

Step 4.3.2.7.2

Subtract $8 \sqrt{5}$ from $9 \sqrt{5}$ .

$\frac{5 + \sqrt{5}}{2} - 9 - 3 \sqrt{5}$

Step 4.3.2.8

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5 + \sqrt{5}}{2} - 9 \cdot \frac{2}{2} - 3 \sqrt{5}$

Step 4.3.2.9

Combine $- 9$ and $\frac{2}{2}$ .

$\frac{5 + \sqrt{5}}{2} + \frac{- 9 \cdot 2}{2} - 3 \sqrt{5}$

Step 4.3.2.10

Simplify the expression.

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Step 4.3.2.10.1

Combine the numerators over the common denominator.

$\frac{5 + \sqrt{5} - 9 \cdot 2}{2} - 3 \sqrt{5}$

Step 4.3.2.10.2

Multiply $- 9$ by $2$ .

$\frac{5 + \sqrt{5} - 18}{2} - 3 \sqrt{5}$

Step 4.3.2.10.3

Subtract $18$ from $5$ .

$\frac{- 13 + \sqrt{5}}{2} - 3 \sqrt{5}$

Step 4.3.2.11

To write $- 3 \sqrt{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- 13 + \sqrt{5}}{2} - 3 \sqrt{5} \cdot \frac{2}{2}$

Step 4.3.2.12

Combine fractions.

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Step 4.3.2.12.1

Combine $- 3 \sqrt{5}$ and $\frac{2}{2}$ .

$\frac{- 13 + \sqrt{5}}{2} + \frac{- 3 \sqrt{5} \cdot 2}{2}$

Step 4.3.2.12.2

Combine the numerators over the common denominator.

$\frac{- 13 + \sqrt{5} - 3 \sqrt{5} \cdot 2}{2}$

Step 4.3.2.13

Simplify the numerator.

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Step 4.3.2.13.1

Multiply $2$ by $- 3$ .

$\frac{- 13 + \sqrt{5} - 6 \sqrt{5}}{2}$

Step 4.3.2.13.2

Subtract $6 \sqrt{5}$ from $\sqrt{5}$ .

$\frac{- 13 - 5 \sqrt{5}}{2}$

Step 4.3.2.14

Simplify with factoring out.

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Step 4.3.2.14.1

Rewrite $- 13$ as $- 1 (13)$ .

$\frac{- 1 (13) - 5 \sqrt{5}}{2}$

Step 4.3.2.14.2

Factor $- 1$ out of $- 5 \sqrt{5}$ .

$\frac{- 1 (13) - (5 \sqrt{5})}{2}$

Step 4.3.2.14.3

Factor $- 1$ out of $- 1 (13) - (5 \sqrt{5})$ .

$\frac{- 1 (13 + 5 \sqrt{5})}{2}$

Step 4.3.2.14.4

Move the negative in front of the fraction.

$- \frac{13 + 5 \sqrt{5}}{2}$

Step 4.4

List all of the points.

$(0, 0), (- \frac{1 - \sqrt{5}}{2}, - \frac{13 - 5 \sqrt{5}}{2}), (- \frac{1 + \sqrt{5}}{2}, - \frac{13 + 5 \sqrt{5}}{2})$

Step 5