Precalculus Examples

$y = \frac{1}{2} x^{3} + x^{2} + 10$ , $y = 5 x^{2} - \frac{13}{2} x + 10$

Step 1

Eliminate the equal sides of each equation and combine.

$\frac{1}{2} x^{3} + x^{2} + 10 = 5 x^{2} - \frac{13}{2} x + 10$

Step 2

Solve $\frac{1}{2} x^{3} + x^{2} + 10 = 5 x^{2} - \frac{13}{2} x + 10$ for $x$ .

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Step 2.1

Combine $\frac{1}{2}$ and $x^{3}$ .

$\frac{x^{3}}{2} + x^{2} + 10 = 5 x^{2} - \frac{13}{2} x + 10$

Step 2.2

Simplify each term.

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Step 2.2.1

Combine $x$ and $\frac{13}{2}$ .

$\frac{x^{3}}{2} + x^{2} + 10 = 5 x^{2} - \frac{x \cdot 13}{2} + 10$

Step 2.2.2

Move $13$ to the left of $x$ .

$\frac{x^{3}}{2} + x^{2} + 10 = 5 x^{2} - \frac{13 x}{2} + 10$

Step 2.3

Move all terms containing $x$ to the left side of the equation.

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Step 2.3.1

Subtract $5 x^{2}$ from both sides of the equation.

$\frac{x^{3}}{2} + x^{2} + 10 - 5 x^{2} = - \frac{13 x}{2} + 10$

Step 2.3.2

Add $\frac{13 x}{2}$ to both sides of the equation.

$\frac{x^{3}}{2} + x^{2} + 10 - 5 x^{2} + \frac{13 x}{2} = 10$

Step 2.3.3

Subtract $5 x^{2}$ from $x^{2}$ .

$\frac{x^{3}}{2} - 4 x^{2} + 10 + \frac{13 x}{2} = 10$

Step 2.4

Multiply each term in $\frac{x^{3}}{2} - 4 x^{2} + 10 + \frac{13 x}{2} = 10$ by $2$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $\frac{x^{3}}{2} - 4 x^{2} + 10 + \frac{13 x}{2} = 10$ by $2$ .

$\frac{x^{3}}{2} \cdot 2 - 4 x^{2} \cdot 2 + 10 \cdot 2 + \frac{13 x}{2} \cdot 2 = 10 \cdot 2$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Cancel the common factor of $2$ .

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Step 2.4.2.1.1.1

Cancel the common factor.

$\frac{x^{3}}{2} \cdot 2 - 4 x^{2} \cdot 2 + 10 \cdot 2 + \frac{13 x}{2} \cdot 2 = 10 \cdot 2$

Step 2.4.2.1.1.2

Rewrite the expression.

$x^{3} - 4 x^{2} \cdot 2 + 10 \cdot 2 + \frac{13 x}{2} \cdot 2 = 10 \cdot 2$

Step 2.4.2.1.2

Multiply $2$ by $- 4$ .

$x^{3} - 8 x^{2} + 10 \cdot 2 + \frac{13 x}{2} \cdot 2 = 10 \cdot 2$

Step 2.4.2.1.3

Multiply $10$ by $2$ .

$x^{3} - 8 x^{2} + 20 + \frac{13 x}{2} \cdot 2 = 10 \cdot 2$

Step 2.4.2.1.4

Cancel the common factor of $2$ .

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Step 2.4.2.1.4.1

Cancel the common factor.

$x^{3} - 8 x^{2} + 20 + \frac{13 x}{2} \cdot 2 = 10 \cdot 2$

Step 2.4.2.1.4.2

Rewrite the expression.

$x^{3} - 8 x^{2} + 20 + 13 x = 10 \cdot 2$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Multiply $10$ by $2$ .

$x^{3} - 8 x^{2} + 20 + 13 x = 20$

Step 2.5

Subtract $20$ from both sides of the equation.

$x^{3} - 8 x^{2} + 20 + 13 x - 20 = 0$

Step 2.6

Combine the opposite terms in $x^{3} - 8 x^{2} + 20 + 13 x - 20$ .

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Step 2.6.1

Subtract $20$ from $20$ .

$x^{3} - 8 x^{2} + 13 x + 0 = 0$

Step 2.6.2

Add $x^{3} - 8 x^{2} + 13 x$ and $0$ .

$x^{3} - 8 x^{2} + 13 x = 0$

Step 2.7

Factor $x$ out of $x^{3} - 8 x^{2} + 13 x$ .

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Step 2.7.1

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

$x \cdot x^{2} - 8 x^{2} + 13 x = 0$

Step 2.7.2

Factor $x$ out of $- 8 x^{2}$ .

$x \cdot x^{2} + x (- 8 x) + 13 x = 0$

Step 2.7.3

Factor $x$ out of $13 x$ .

$x \cdot x^{2} + x (- 8 x) + x \cdot 13 = 0$

Step 2.7.4

Factor $x$ out of $x \cdot x^{2} + x (- 8 x)$ .

$x (x^{2} - 8 x) + x \cdot 13 = 0$

Step 2.7.5

Factor $x$ out of $x (x^{2} - 8 x) + x \cdot 13$ .

$x (x^{2} - 8 x + 13) = 0$

Step 2.8

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

$x = 0$

$x^{2} - 8 x + 13 = 0 + y = 5 x^{2} - \frac{13}{2} x + 10$

Step 2.9

Set $x$ equal to $0$ .

$x = 0$

Step 2.10

Set $x^{2} - 8 x + 13$ equal to $0$ and solve for $x$ .

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Step 2.10.1

Set $x^{2} - 8 x + 13$ equal to $0$ .

$x^{2} - 8 x + 13 = 0$

Step 2.10.2

Solve $x^{2} - 8 x + 13 = 0$ for $x$ .

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Step 2.10.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = 5 x^{2} - \frac{13}{2} x + 10$

Step 2.10.2.2

Substitute the values $a = 1$ , $b = - 8$ , and $c = 13$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot 13)}}{2 \cdot 1} + y = 5 x^{2} - \frac{13}{2} x + 10$

Step 2.10.2.3

Simplify.

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Step 2.10.2.3.1

Simplify the numerator.

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Step 2.10.2.3.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Step 2.10.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 13$ .

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Step 2.10.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 13}}{2 \cdot 1}$

Step 2.10.2.3.1.2.2

Multiply $- 4$ by $13$ .

$x = \frac{8 \pm \sqrt{64 - 52}}{2 \cdot 1}$

Step 2.10.2.3.1.3

Subtract $52$ from $64$ .

$x = \frac{8 \pm \sqrt{12}}{2 \cdot 1}$

Step 2.10.2.3.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.10.2.3.1.4.1

Factor $4$ out of $12$ .

$x = \frac{8 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 2.10.2.3.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{8 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 2.10.2.3.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 2.10.2.3.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{3}}{2}$

Step 2.10.2.3.3

Simplify $\frac{8 \pm 2 \sqrt{3}}{2}$ .

$x = 4 \pm \sqrt{3}$

Step 2.10.2.4

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

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Step 2.10.2.4.1

Simplify the numerator.

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Step 2.10.2.4.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Step 2.10.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 13$ .

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Step 2.10.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 13}}{2 \cdot 1}$

Step 2.10.2.4.1.2.2

Multiply $- 4$ by $13$ .

$x = \frac{8 \pm \sqrt{64 - 52}}{2 \cdot 1}$

Step 2.10.2.4.1.3

Subtract $52$ from $64$ .

$x = \frac{8 \pm \sqrt{12}}{2 \cdot 1}$

Step 2.10.2.4.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.10.2.4.1.4.1

Factor $4$ out of $12$ .

$x = \frac{8 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 2.10.2.4.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{8 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 2.10.2.4.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 2.10.2.4.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{3}}{2}$

Step 2.10.2.4.3

Simplify $\frac{8 \pm 2 \sqrt{3}}{2}$ .

$x = 4 \pm \sqrt{3}$

Step 2.10.2.4.4

Change the $\pm$ to $+$ .

$x = 4 + \sqrt{3}$

Step 2.10.2.5

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

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Step 2.10.2.5.1

Simplify the numerator.

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Step 2.10.2.5.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Step 2.10.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 13$ .

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Step 2.10.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 13}}{2 \cdot 1}$

Step 2.10.2.5.1.2.2

Multiply $- 4$ by $13$ .

$x = \frac{8 \pm \sqrt{64 - 52}}{2 \cdot 1}$

Step 2.10.2.5.1.3

Subtract $52$ from $64$ .

$x = \frac{8 \pm \sqrt{12}}{2 \cdot 1}$

Step 2.10.2.5.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.10.2.5.1.4.1

Factor $4$ out of $12$ .

$x = \frac{8 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 2.10.2.5.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{8 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 2.10.2.5.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 2.10.2.5.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{3}}{2}$

Step 2.10.2.5.3

Simplify $\frac{8 \pm 2 \sqrt{3}}{2}$ .

$x = 4 \pm \sqrt{3}$

Step 2.10.2.5.4

Change the $\pm$ to $-$ .

$x = 4 - \sqrt{3}$

Step 2.10.2.6

The final answer is the combination of both solutions.

$x = 4 + \sqrt{3}, 4 - \sqrt{3}$

Step 2.11

The final solution is all the values that make $x (x^{2} - 8 x + 13) = 0$ true.

$x = 0, 4 + \sqrt{3}, 4 - \sqrt{3}$

Step 3

Evaluate $y$ when $x = 0$ .

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Step 3.1

Substitute $0$ for $x$ .

$y = 5 {(0)}^{2} - \frac{13}{2} \cdot 0 + 10$

Step 3.2

Substitute $0$ for $x$ in $y = 5 {(0)}^{2} - \frac{13}{2} \cdot 0 + 10$ and solve for $y$ .

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Step 3.2.1

Remove parentheses.

$y = 5 \cdot 0^{2} - \frac{13}{2} \cdot 0 + 10$

Step 3.2.2

Remove parentheses.

$y = 5 {(0)}^{2} - \frac{13}{2} \cdot 0 + 10$

Step 3.2.3

Simplify $5 {(0)}^{2} - \frac{13}{2} \cdot 0 + 10$ .

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

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

$y = 5 \cdot 0 - \frac{13}{2} \cdot 0 + 10$

Step 3.2.3.1.2

Multiply $5$ by $0$ .

$y = 0 - \frac{13}{2} \cdot 0 + 10$

Step 3.2.3.1.3

Multiply $- \frac{13}{2} \cdot 0$ .

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Step 3.2.3.1.3.1

Multiply $0$ by $- 1$ .

$y = 0 + 0 (\frac{13}{2}) + 10$

Step 3.2.3.1.3.2

Multiply $0$ by $\frac{13}{2}$ .

$y = 0 + 0 + 10$

Step 3.2.3.2

Simplify by adding numbers.

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Step 3.2.3.2.1

Add $0$ and $0$ .

$y = 0 + 10$

Step 3.2.3.2.2

Add $0$ and $10$ .

$y = 10$

Step 4

Evaluate $y$ when $x = 4 + \sqrt{3}$ .

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Step 4.1

Substitute $4 + \sqrt{3}$ for $x$ .

$y = 5 {(4 + \sqrt{3})}^{2} - \frac{13}{2} \cdot (4 + \sqrt{3}) + 10$

Step 4.2

Simplify $5 {(4 + \sqrt{3})}^{2} - \frac{13}{2} \cdot (4 + \sqrt{3}) + 10$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Rewrite ${(4 + \sqrt{3})}^{2}$ as $(4 + \sqrt{3}) (4 + \sqrt{3})$ .

$y = 5 ((4 + \sqrt{3}) (4 + \sqrt{3})) - \frac{13}{2} \cdot (4 + \sqrt{3}) + 10$

Step 4.2.1.2

Expand $(4 + \sqrt{3}) (4 + \sqrt{3})$ using the FOIL Method.

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Step 4.2.1.2.1

Apply the distributive property.

$y = 5 (4 (4 + \sqrt{3}) + \sqrt{3} (4 + \sqrt{3})) - \frac{13}{2} \cdot (4 + \sqrt{3}) + 10$

Step 4.2.1.2.2

Apply the distributive property.

$y = 5 (4 \cdot 4 + 4 \sqrt{3} + \sqrt{3} (4 + \sqrt{3})) - \frac{13}{2} \cdot (4 + \sqrt{3}) + 10$

Step 4.2.1.2.3

Apply the distributive property.

$y = 5 (4 \cdot 4 + 4 \sqrt{3} + \sqrt{3} \cdot 4 + \sqrt{3} \sqrt{3}) - \frac{13}{2} \cdot (4 + \sqrt{3}) + 10$

Step 4.2.1.3

Simplify and combine like terms.

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Step 4.2.1.3.1

Simplify each term.

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Step 4.2.1.3.1.1

Multiply $4$ by $4$ .

$y = 5 (16 + 4 \sqrt{3} + \sqrt{3} \cdot 4 + \sqrt{3} \sqrt{3}) - \frac{13}{2} \cdot (4 + \sqrt{3}) + 10$

Step 4.2.1.3.1.2

Move $4$ to the left of $\sqrt{3}$ .

$y = 5 (16 + 4 \sqrt{3} + 4 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}) - \frac{13}{2} \cdot (4 + \sqrt{3}) + 10$

Step 4.2.1.3.1.3

Combine using the product rule for radicals.

$y = 5 (16 + 4 \sqrt{3} + 4 \sqrt{3} + \sqrt{3 \cdot 3}) - \frac{13}{2} \cdot (4 + \sqrt{3}) + 10$

Step 4.2.1.3.1.4

Multiply $3$ by $3$ .

$y = 5 (16 + 4 \sqrt{3} + 4 \sqrt{3} + \sqrt{9}) - \frac{13}{2} \cdot (4 + \sqrt{3}) + 10$

Step 4.2.1.3.1.5

Rewrite $9$ as $3^{2}$ .

$y = 5 (16 + 4 \sqrt{3} + 4 \sqrt{3} + \sqrt{3^{2}}) - \frac{13}{2} \cdot (4 + \sqrt{3}) + 10$

Step 4.2.1.3.1.6

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

$y = 5 (16 + 4 \sqrt{3} + 4 \sqrt{3} + 3) - \frac{13}{2} \cdot (4 + \sqrt{3}) + 10$

Step 4.2.1.3.2

Add $16$ and $3$ .

$y = 5 (19 + 4 \sqrt{3} + 4 \sqrt{3}) - \frac{13}{2} \cdot (4 + \sqrt{3}) + 10$

Step 4.2.1.3.3

Add $4 \sqrt{3}$ and $4 \sqrt{3}$ .

$y = 5 (19 + 8 \sqrt{3}) - \frac{13}{2} \cdot (4 + \sqrt{3}) + 10$

Step 4.2.1.4

Apply the distributive property.

$y = 5 \cdot 19 + 5 (8 \sqrt{3}) - \frac{13}{2} \cdot (4 + \sqrt{3}) + 10$

Step 4.2.1.5

Multiply $5$ by $19$ .

$y = 95 + 5 (8 \sqrt{3}) - \frac{13}{2} \cdot (4 + \sqrt{3}) + 10$

Step 4.2.1.6

Multiply $8$ by $5$ .

$y = 95 + 40 \sqrt{3} - \frac{13}{2} \cdot (4 + \sqrt{3}) + 10$

Step 4.2.1.7

Apply the distributive property.

$y = 95 + 40 \sqrt{3} - \frac{13}{2} \cdot 4 - \frac{13}{2} \sqrt{3} + 10$

Step 4.2.1.8

Cancel the common factor of $2$ .

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Step 4.2.1.8.1

Move the leading negative in $- \frac{13}{2}$ into the numerator.

$y = 95 + 40 \sqrt{3} + \frac{- 13}{2} \cdot 4 - \frac{13}{2} \sqrt{3} + 10$

Step 4.2.1.8.2

Factor $2$ out of $4$ .

$y = 95 + 40 \sqrt{3} + \frac{- 13}{2} \cdot (2 (2)) - \frac{13}{2} \sqrt{3} + 10$

Step 4.2.1.8.3

Cancel the common factor.

$y = 95 + 40 \sqrt{3} + \frac{- 13}{2} \cdot (2 \cdot 2) - \frac{13}{2} \sqrt{3} + 10$

Step 4.2.1.8.4

Rewrite the expression.

$y = 95 + 40 \sqrt{3} - 13 \cdot 2 - \frac{13}{2} \sqrt{3} + 10$

Step 4.2.1.9

Multiply $- 13$ by $2$ .

$y = 95 + 40 \sqrt{3} - 26 - \frac{13}{2} \sqrt{3} + 10$

Step 4.2.1.10

Combine $\sqrt{3}$ and $\frac{13}{2}$ .

$y = 95 + 40 \sqrt{3} - 26 - \frac{\sqrt{3} \cdot 13}{2} + 10$

Step 4.2.1.11

Move $13$ to the left of $\sqrt{3}$ .

$y = 95 + 40 \sqrt{3} - 26 - \frac{13 \sqrt{3}}{2} + 10$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Subtract $26$ from $95$ .

$y = 69 + 40 \sqrt{3} - \frac{13 \sqrt{3}}{2} + 10$

Step 4.2.2.2

Add $69$ and $10$ .

$y = 79 + 40 \sqrt{3} - \frac{13 \sqrt{3}}{2}$

Step 4.2.3

To write $40 \sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = 79 + 40 \sqrt{3} \cdot \frac{2}{2} - \frac{13 \sqrt{3}}{2}$

Step 4.2.4

Combine fractions.

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Step 4.2.4.1

Combine $40 \sqrt{3}$ and $\frac{2}{2}$ .

$y = 79 + \frac{40 \sqrt{3} \cdot 2}{2} - \frac{13 \sqrt{3}}{2}$

Step 4.2.4.2

Combine the numerators over the common denominator.

$y = 79 + \frac{40 \sqrt{3} \cdot 2 - 13 \sqrt{3}}{2}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $2$ by $40$ .

$y = 79 + \frac{80 \sqrt{3} - 13 \sqrt{3}}{2}$

Step 4.2.5.2

Subtract $13 \sqrt{3}$ from $80 \sqrt{3}$ .

$y = 79 + \frac{67 \sqrt{3}}{2}$

Step 5

Evaluate $y$ when $x = 4 - \sqrt{3}$ .

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Step 5.1

Substitute $4 - \sqrt{3}$ for $x$ .

$y = 5 {(4 - \sqrt{3})}^{2} - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2

Simplify $5 {(4 - \sqrt{3})}^{2} - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$ .

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Rewrite ${(4 - \sqrt{3})}^{2}$ as $(4 - \sqrt{3}) (4 - \sqrt{3})$ .

$y = 5 ((4 - \sqrt{3}) (4 - \sqrt{3})) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.2

Expand $(4 - \sqrt{3}) (4 - \sqrt{3})$ using the FOIL Method.

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Step 5.2.1.2.1

Apply the distributive property.

$y = 5 (4 (4 - \sqrt{3}) - \sqrt{3} (4 - \sqrt{3})) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.2.2

Apply the distributive property.

$y = 5 (4 \cdot 4 + 4 (- \sqrt{3}) - \sqrt{3} (4 - \sqrt{3})) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.2.3

Apply the distributive property.

$y = 5 (4 \cdot 4 + 4 (- \sqrt{3}) - \sqrt{3} \cdot 4 - \sqrt{3} (- \sqrt{3})) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.3

Simplify and combine like terms.

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Step 5.2.1.3.1

Simplify each term.

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Step 5.2.1.3.1.1

Multiply $4$ by $4$ .

$y = 5 (16 + 4 (- \sqrt{3}) - \sqrt{3} \cdot 4 - \sqrt{3} (- \sqrt{3})) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.3.1.2

Multiply $- 1$ by $4$ .

$y = 5 (16 - 4 \sqrt{3} - \sqrt{3} \cdot 4 - \sqrt{3} (- \sqrt{3})) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.3.1.3

Multiply $4$ by $- 1$ .

$y = 5 (16 - 4 \sqrt{3} - 4 \sqrt{3} - \sqrt{3} (- \sqrt{3})) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.3.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

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Step 5.2.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$y = 5 (16 - 4 \sqrt{3} - 4 \sqrt{3} + 1 \sqrt{3} \sqrt{3}) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.3.1.4.2

Multiply $\sqrt{3}$ by $1$ .

$y = 5 (16 - 4 \sqrt{3} - 4 \sqrt{3} + \sqrt{3} \sqrt{3}) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.3.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$y = 5 (16 - 4 \sqrt{3} - 4 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.3.1.4.4

Raise $\sqrt{3}$ to the power of $1$ .

$y = 5 (16 - 4 \sqrt{3} - 4 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.3.1.4.5

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

$y = 5 (16 - 4 \sqrt{3} - 4 \sqrt{3} + {\sqrt{3}}^{1 + 1}) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.3.1.4.6

Add $1$ and $1$ .

$y = 5 (16 - 4 \sqrt{3} - 4 \sqrt{3} + {\sqrt{3}}^{2}) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.3.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.2.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y = 5 (16 - 4 \sqrt{3} - 4 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.3.1.5.2

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

$y = 5 (16 - 4 \sqrt{3} - 4 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.3.1.5.3

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

$y = 5 (16 - 4 \sqrt{3} - 4 \sqrt{3} + 3^{\frac{2}{2}}) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 5.2.1.3.1.5.4.1

Cancel the common factor.

$y = 5 (16 - 4 \sqrt{3} - 4 \sqrt{3} + 3^{\frac{2}{2}}) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.3.1.5.4.2

Rewrite the expression.

$y = 5 (16 - 4 \sqrt{3} - 4 \sqrt{3} + 3^{1}) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.3.1.5.5

Evaluate the exponent.

$y = 5 (16 - 4 \sqrt{3} - 4 \sqrt{3} + 3) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.3.2

Add $16$ and $3$ .

$y = 5 (19 - 4 \sqrt{3} - 4 \sqrt{3}) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.3.3

Subtract $4 \sqrt{3}$ from $- 4 \sqrt{3}$ .

$y = 5 (19 - 8 \sqrt{3}) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.4

Apply the distributive property.

$y = 5 \cdot 19 + 5 (- 8 \sqrt{3}) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.5

Multiply $5$ by $19$ .

$y = 95 + 5 (- 8 \sqrt{3}) - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.6

Multiply $- 8$ by $5$ .

$y = 95 - 40 \sqrt{3} - \frac{13}{2} \cdot (4 - \sqrt{3}) + 10$

Step 5.2.1.7

Apply the distributive property.

$y = 95 - 40 \sqrt{3} - \frac{13}{2} \cdot 4 - \frac{13}{2} (- \sqrt{3}) + 10$

Step 5.2.1.8

Cancel the common factor of $2$ .

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Step 5.2.1.8.1

Move the leading negative in $- \frac{13}{2}$ into the numerator.

$y = 95 - 40 \sqrt{3} + \frac{- 13}{2} \cdot 4 - \frac{13}{2} (- \sqrt{3}) + 10$

Step 5.2.1.8.2

Factor $2$ out of $4$ .

$y = 95 - 40 \sqrt{3} + \frac{- 13}{2} \cdot (2 (2)) - \frac{13}{2} (- \sqrt{3}) + 10$

Step 5.2.1.8.3

Cancel the common factor.

$y = 95 - 40 \sqrt{3} + \frac{- 13}{2} \cdot (2 \cdot 2) - \frac{13}{2} (- \sqrt{3}) + 10$

Step 5.2.1.8.4

Rewrite the expression.

$y = 95 - 40 \sqrt{3} - 13 \cdot 2 - \frac{13}{2} (- \sqrt{3}) + 10$

Step 5.2.1.9

Multiply $- 13$ by $2$ .

$y = 95 - 40 \sqrt{3} - 26 - \frac{13}{2} (- \sqrt{3}) + 10$

Step 5.2.1.10

Multiply $- \frac{13}{2} (- \sqrt{3})$ .

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Step 5.2.1.10.1

Multiply $- 1$ by $- 1$ .

$y = 95 - 40 \sqrt{3} - 26 + 1 (\frac{13}{2}) \sqrt{3} + 10$

Step 5.2.1.10.2

Multiply $\frac{13}{2}$ by $1$ .

$y = 95 - 40 \sqrt{3} - 26 + \frac{13}{2} \sqrt{3} + 10$

Step 5.2.1.10.3

Combine $\frac{13}{2}$ and $\sqrt{3}$ .

$y = 95 - 40 \sqrt{3} - 26 + \frac{13 \sqrt{3}}{2} + 10$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $26$ from $95$ .

$y = 69 - 40 \sqrt{3} + \frac{13 \sqrt{3}}{2} + 10$

Step 5.2.2.2

Add $69$ and $10$ .

$y = 79 - 40 \sqrt{3} + \frac{13 \sqrt{3}}{2}$

Step 5.2.3

To write $- 40 \sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = 79 - 40 \sqrt{3} \cdot \frac{2}{2} + \frac{13 \sqrt{3}}{2}$

Step 5.2.4

Combine fractions.

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Step 5.2.4.1

Combine $- 40 \sqrt{3}$ and $\frac{2}{2}$ .

$y = 79 + \frac{- 40 \sqrt{3} \cdot 2}{2} + \frac{13 \sqrt{3}}{2}$

Step 5.2.4.2

Combine the numerators over the common denominator.

$y = 79 + \frac{- 40 \sqrt{3} \cdot 2 + 13 \sqrt{3}}{2}$

Step 5.2.5

Simplify each term.

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Step 5.2.5.1

Simplify the numerator.

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Step 5.2.5.1.1

Multiply $2$ by $- 40$ .

$y = 79 + \frac{- 80 \sqrt{3} + 13 \sqrt{3}}{2}$

Step 5.2.5.1.2

Add $- 80 \sqrt{3}$ and $13 \sqrt{3}$ .

$y = 79 + \frac{- 67 \sqrt{3}}{2}$

Step 5.2.5.2

Move the negative in front of the fraction.

$y = 79 - \frac{67 \sqrt{3}}{2}$

Step 6

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(0, 10)$

$(4 + \sqrt{3}, 79 + \frac{67 \sqrt{3}}{2})$

$(4 - \sqrt{3}, 79 - \frac{67 \sqrt{3}}{2})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(0, 10), (4 + \sqrt{3}, 79 + \frac{67 \sqrt{3}}{2}), (4 - \sqrt{3}, 79 - \frac{67 \sqrt{3}}{2})$

Equation Form:

$x = 0, y = 10$

$x = 4 + \sqrt{3}, y = 79 + \frac{67 \sqrt{3}}{2}$

$x = 4 - \sqrt{3}, y = 79 - \frac{67 \sqrt{3}}{2}$

Step 8