Calculus Examples

$f (x) = (x - 1) (x - 2) (x - 3)$

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

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = (x - 1) (x - 2)$ and $g (x) = x - 3$ .

$(x - 1) (x - 2) \frac{d}{d x} [x - 3] + (x - 3) \frac{d}{d x} [(x - 1) (x - 2)]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

$(x - 1) (x - 2) (\frac{d}{d x} [x] + \frac{d}{d x} [- 3]) + (x - 3) \frac{d}{d x} [(x - 1) (x - 2)]$

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 $n = 1$ .

$(x - 1) (x - 2) (1 + \frac{d}{d x} [- 3]) + (x - 3) \frac{d}{d x} [(x - 1) (x - 2)]$

Step 1.1.2.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$(x - 1) (x - 2) (1 + 0) + (x - 3) \frac{d}{d x} [(x - 1) (x - 2)]$

Step 1.1.2.4

Simplify the expression.

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Step 1.1.2.4.1

Add $1$ and $0$ .

$(x - 1) (x - 2) \cdot 1 + (x - 3) \frac{d}{d x} [(x - 1) (x - 2)]$

Step 1.1.2.4.2

Multiply $x - 1$ by $1$ .

$(x - 1) (x - 2) + (x - 3) \frac{d}{d x} [(x - 1) (x - 2)]$

Step 1.1.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x - 1$ and $g (x) = x - 2$ .

$(x - 1) (x - 2) + (x - 3) ((x - 1) \frac{d}{d x} [x - 2] + (x - 2) \frac{d}{d x} [x - 1])$

Step 1.1.4

Differentiate.

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Step 1.1.4.1

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

$(x - 1) (x - 2) + (x - 3) ((x - 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 2]) + (x - 2) \frac{d}{d x} [x - 1])$

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 $n = 1$ .

$(x - 1) (x - 2) + (x - 3) ((x - 1) (1 + \frac{d}{d x} [- 2]) + (x - 2) \frac{d}{d x} [x - 1])$

Step 1.1.4.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$(x - 1) (x - 2) + (x - 3) ((x - 1) (1 + 0) + (x - 2) \frac{d}{d x} [x - 1])$

Step 1.1.4.4

Simplify the expression.

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Step 1.1.4.4.1

Add $1$ and $0$ .

$(x - 1) (x - 2) + (x - 3) ((x - 1) \cdot 1 + (x - 2) \frac{d}{d x} [x - 1])$

Step 1.1.4.4.2

Multiply $x - 1$ by $1$ .

$(x - 1) (x - 2) + (x - 3) (x - 1 + (x - 2) \frac{d}{d x} [x - 1])$

Step 1.1.4.5

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$(x - 1) (x - 2) + (x - 3) (x - 1 + (x - 2) (\frac{d}{d x} [x] + \frac{d}{d x} [- 1]))$

Step 1.1.4.6

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

$(x - 1) (x - 2) + (x - 3) (x - 1 + (x - 2) (1 + \frac{d}{d x} [- 1]))$

Step 1.1.4.7

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$(x - 1) (x - 2) + (x - 3) (x - 1 + (x - 2) (1 + 0))$

Step 1.1.4.8

Simplify by adding terms.

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Step 1.1.4.8.1

Add $1$ and $0$ .

$(x - 1) (x - 2) + (x - 3) (x - 1 + (x - 2) \cdot 1)$

Step 1.1.4.8.2

Multiply $x - 2$ by $1$ .

$(x - 1) (x - 2) + (x - 3) (x - 1 + x - 2)$

Step 1.1.4.8.3

Add $x$ and $x$ .

$(x - 1) (x - 2) + (x - 3) (2 x - 1 - 2)$

Step 1.1.4.8.4

Subtract $2$ from $- 1$ .

$(x - 1) (x - 2) + (x - 3) (2 x - 3)$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Apply the distributive property.

$(x - 1) x + (x - 1) \cdot - 2 + (x - 3) (2 x - 3)$

Step 1.1.5.2

Apply the distributive property.

$x \cdot x - 1 x + (x - 1) \cdot - 2 + (x - 3) (2 x - 3)$

Step 1.1.5.3

Apply the distributive property.

$x \cdot x - 1 x + x \cdot - 2 - 1 \cdot - 2 + (x - 3) (2 x - 3)$

Step 1.1.5.4

Apply the distributive property.

$x \cdot x - 1 x + x \cdot - 2 - 1 \cdot - 2 + (x - 3) (2 x) + (x - 3) \cdot - 3$

Step 1.1.5.5

Apply the distributive property.

$x \cdot x - 1 x + x \cdot - 2 - 1 \cdot - 2 + x (2 x) - 3 (2 x) + (x - 3) \cdot - 3$

Step 1.1.5.6

Apply the distributive property.

$x \cdot x - 1 x + x \cdot - 2 - 1 \cdot - 2 + x (2 x) - 3 (2 x) + x \cdot - 3 - 3 \cdot - 3$

Step 1.1.5.7

Combine terms.

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Step 1.1.5.7.1

Raise $x$ to the power of $1$ .

$x^{1} x - 1 x + x \cdot - 2 - 1 \cdot - 2 + x (2 x) - 3 (2 x) + x \cdot - 3 - 3 \cdot - 3$

Step 1.1.5.7.2

Raise $x$ to the power of $1$ .

$x^{1} x^{1} - 1 x + x \cdot - 2 - 1 \cdot - 2 + x (2 x) - 3 (2 x) + x \cdot - 3 - 3 \cdot - 3$

Step 1.1.5.7.3

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

$x^{1 + 1} - 1 x + x \cdot - 2 - 1 \cdot - 2 + x (2 x) - 3 (2 x) + x \cdot - 3 - 3 \cdot - 3$

Step 1.1.5.7.4

Add $1$ and $1$ .

$x^{2} - 1 x + x \cdot - 2 - 1 \cdot - 2 + x (2 x) - 3 (2 x) + x \cdot - 3 - 3 \cdot - 3$

Step 1.1.5.7.5

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x + x \cdot - 2 - 1 \cdot - 2 + x (2 x) - 3 (2 x) + x \cdot - 3 - 3 \cdot - 3$

Step 1.1.5.7.6

Move $- 2$ to the left of $x$ .

$x^{2} - x - 2 \cdot x - 1 \cdot - 2 + x (2 x) - 3 (2 x) + x \cdot - 3 - 3 \cdot - 3$

Step 1.1.5.7.7

Multiply $- 1$ by $- 2$ .

$x^{2} - x - 2 x + 2 + x (2 x) - 3 (2 x) + x \cdot - 3 - 3 \cdot - 3$

Step 1.1.5.7.8

Subtract $2 x$ from $- x$ .

$x^{2} - 3 x + 2 + x (2 x) - 3 (2 x) + x \cdot - 3 - 3 \cdot - 3$

Step 1.1.5.7.9

Raise $x$ to the power of $1$ .

$x^{2} - 3 x + 2 + 2 (x^{1} x) - 3 (2 x) + x \cdot - 3 - 3 \cdot - 3$

Step 1.1.5.7.10

Raise $x$ to the power of $1$ .

$x^{2} - 3 x + 2 + 2 (x^{1} x^{1}) - 3 (2 x) + x \cdot - 3 - 3 \cdot - 3$

Step 1.1.5.7.11

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

$x^{2} - 3 x + 2 + 2 x^{1 + 1} - 3 (2 x) + x \cdot - 3 - 3 \cdot - 3$

Step 1.1.5.7.12

Add $1$ and $1$ .

$x^{2} - 3 x + 2 + 2 x^{2} - 3 (2 x) + x \cdot - 3 - 3 \cdot - 3$

Step 1.1.5.7.13

Multiply $2$ by $- 3$ .

$x^{2} - 3 x + 2 + 2 x^{2} - 6 x + x \cdot - 3 - 3 \cdot - 3$

Step 1.1.5.7.14

Move $- 3$ to the left of $x$ .

$x^{2} - 3 x + 2 + 2 x^{2} - 6 x - 3 \cdot x - 3 \cdot - 3$

Step 1.1.5.7.15

Multiply $- 3$ by $- 3$ .

$x^{2} - 3 x + 2 + 2 x^{2} - 6 x - 3 x + 9$

Step 1.1.5.7.16

Subtract $3 x$ from $- 6 x$ .

$x^{2} - 3 x + 2 + 2 x^{2} - 9 x + 9$

Step 1.1.5.7.17

Add $x^{2}$ and $2 x^{2}$ .

$3 x^{2} - 3 x + 2 - 9 x + 9$

Step 1.1.5.7.18

Subtract $9 x$ from $- 3 x$ .

$3 x^{2} - 12 x + 2 + 9$

Step 1.1.5.7.19

Add $2$ and $9$ .

$f' (x) = 3 x^{2} - 12 x + 11$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} - 12 x + 11$ .

$3 x^{2} - 12 x + 11$

Step 2

Set the first derivative equal to $0$ then solve the equation $3 x^{2} - 12 x + 11 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$3 x^{2} - 12 x + 11 = 0$

Step 2.2

Use the quadratic formula to find the solutions.

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

Step 2.3

Substitute the values $a = 3$ , $b = - 12$ , and $c = 11$ into the quadratic formula and solve for $x$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (3 \cdot 11)}}{2 \cdot 3}$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 3 \cdot 11}}{2 \cdot 3}$

Step 2.4.1.2

Multiply $- 4 \cdot 3 \cdot 11$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{12 \pm \sqrt{144 - 12 \cdot 11}}{2 \cdot 3}$

Step 2.4.1.2.2

Multiply $- 12$ by $11$ .

$x = \frac{12 \pm \sqrt{144 - 132}}{2 \cdot 3}$

Step 2.4.1.3

Subtract $132$ from $144$ .

$x = \frac{12 \pm \sqrt{12}}{2 \cdot 3}$

Step 2.4.1.4

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

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Step 2.4.1.4.1

Factor $4$ out of $12$ .

$x = \frac{12 \pm \sqrt{4 (3)}}{2 \cdot 3}$

Step 2.4.1.4.2

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

$x = \frac{12 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 3}$

Step 2.4.1.5

Pull terms out from under the radical.

$x = \frac{12 \pm 2 \sqrt{3}}{2 \cdot 3}$

Step 2.4.2

Multiply $2$ by $3$ .

$x = \frac{12 \pm 2 \sqrt{3}}{6}$

Step 2.4.3

Simplify $\frac{12 \pm 2 \sqrt{3}}{6}$ .

$x = \frac{6 \pm \sqrt{3}}{3}$

Step 2.5

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

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 3 \cdot 11}}{2 \cdot 3}$

Step 2.5.1.2

Multiply $- 4 \cdot 3 \cdot 11$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{12 \pm \sqrt{144 - 12 \cdot 11}}{2 \cdot 3}$

Step 2.5.1.2.2

Multiply $- 12$ by $11$ .

$x = \frac{12 \pm \sqrt{144 - 132}}{2 \cdot 3}$

Step 2.5.1.3

Subtract $132$ from $144$ .

$x = \frac{12 \pm \sqrt{12}}{2 \cdot 3}$

Step 2.5.1.4

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

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Step 2.5.1.4.1

Factor $4$ out of $12$ .

$x = \frac{12 \pm \sqrt{4 (3)}}{2 \cdot 3}$

Step 2.5.1.4.2

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

$x = \frac{12 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 3}$

Step 2.5.1.5

Pull terms out from under the radical.

$x = \frac{12 \pm 2 \sqrt{3}}{2 \cdot 3}$

Step 2.5.2

Multiply $2$ by $3$ .

$x = \frac{12 \pm 2 \sqrt{3}}{6}$

Step 2.5.3

Simplify $\frac{12 \pm 2 \sqrt{3}}{6}$ .

$x = \frac{6 \pm \sqrt{3}}{3}$

Step 2.5.4

Change the $\pm$ to $+$ .

$x = \frac{6 + \sqrt{3}}{3}$

Step 2.6

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

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 3 \cdot 11}}{2 \cdot 3}$

Step 2.6.1.2

Multiply $- 4 \cdot 3 \cdot 11$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{12 \pm \sqrt{144 - 12 \cdot 11}}{2 \cdot 3}$

Step 2.6.1.2.2

Multiply $- 12$ by $11$ .

$x = \frac{12 \pm \sqrt{144 - 132}}{2 \cdot 3}$

Step 2.6.1.3

Subtract $132$ from $144$ .

$x = \frac{12 \pm \sqrt{12}}{2 \cdot 3}$

Step 2.6.1.4

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

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Step 2.6.1.4.1

Factor $4$ out of $12$ .

$x = \frac{12 \pm \sqrt{4 (3)}}{2 \cdot 3}$

Step 2.6.1.4.2

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

$x = \frac{12 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 3}$

Step 2.6.1.5

Pull terms out from under the radical.

$x = \frac{12 \pm 2 \sqrt{3}}{2 \cdot 3}$

Step 2.6.2

Multiply $2$ by $3$ .

$x = \frac{12 \pm 2 \sqrt{3}}{6}$

Step 2.6.3

Simplify $\frac{12 \pm 2 \sqrt{3}}{6}$ .

$x = \frac{6 \pm \sqrt{3}}{3}$

Step 2.6.4

Change the $\pm$ to $-$ .

$x = \frac{6 - \sqrt{3}}{3}$

Step 2.7

The final answer is the combination of both solutions.

$x = \frac{6 + \sqrt{3}}{3}, \frac{6 - \sqrt{3}}{3}$

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 $(x - 1) (x - 2) (x - 3)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{6 + \sqrt{3}}{3}$ .

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Step 4.1.1

Substitute $\frac{6 + \sqrt{3}}{3}$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4.1.2.2

Combine fractions.

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Step 4.1.2.2.1

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 4.1.2.2.2

Combine the numerators over the common denominator.

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

Step 4.1.2.3

Simplify the numerator.

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Step 4.1.2.3.1

Multiply $- 1$ by $3$ .

$\frac{6 + \sqrt{3} - 3}{3} ((\frac{6 + \sqrt{3}}{3}) - 2) ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.3.2

Subtract $3$ from $6$ .

$\frac{3 + \sqrt{3}}{3} ((\frac{6 + \sqrt{3}}{3}) - 2) ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.4

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

$\frac{3 + \sqrt{3}}{3} (\frac{6 + \sqrt{3}}{3} - 2 \cdot \frac{3}{3}) ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.5

Combine fractions.

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Step 4.1.2.5.1

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

$\frac{3 + \sqrt{3}}{3} (\frac{6 + \sqrt{3}}{3} + \frac{- 2 \cdot 3}{3}) ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.5.2

Combine the numerators over the common denominator.

$\frac{3 + \sqrt{3}}{3} \cdot \frac{6 + \sqrt{3} - 2 \cdot 3}{3} ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.6

Simplify the numerator.

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Step 4.1.2.6.1

Multiply $- 2$ by $3$ .

$\frac{3 + \sqrt{3}}{3} \cdot \frac{6 + \sqrt{3} - 6}{3} ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.6.2

Subtract $6$ from $6$ .

$\frac{3 + \sqrt{3}}{3} \cdot \frac{0 + \sqrt{3}}{3} ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.6.3

Add $0$ and $\sqrt{3}$ .

$\frac{3 + \sqrt{3}}{3} \cdot \frac{\sqrt{3}}{3} ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.7

Multiply $\frac{3 + \sqrt{3}}{3} \cdot \frac{\sqrt{3}}{3}$ .

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Step 4.1.2.7.1

Multiply $\frac{3 + \sqrt{3}}{3}$ by $\frac{\sqrt{3}}{3}$ .

$\frac{(3 + \sqrt{3}) \sqrt{3}}{3 \cdot 3} ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.7.2

Multiply $3$ by $3$ .

$\frac{(3 + \sqrt{3}) \sqrt{3}}{9} ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.8

Simplify terms.

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Step 4.1.2.8.1

Apply the distributive property.

$\frac{3 \sqrt{3} + \sqrt{3} \sqrt{3}}{9} ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.8.2

Combine using the product rule for radicals.

$\frac{3 \sqrt{3} + \sqrt{3 \cdot 3}}{9} ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.9

Simplify each term.

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Step 4.1.2.9.1

Multiply $3$ by $3$ .

$\frac{3 \sqrt{3} + \sqrt{9}}{9} ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.9.2

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

$\frac{3 \sqrt{3} + \sqrt{3^{2}}}{9} ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.9.3

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

$\frac{3 \sqrt{3} + 3}{9} ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.10

Cancel the common factor of $3 \sqrt{3} + 3$ and $9$ .

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Step 4.1.2.10.1

Factor $3$ out of $3 \sqrt{3}$ .

$\frac{3 (\sqrt{3}) + 3}{9} ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.10.2

Factor $3$ out of $3$ .

$\frac{3 (\sqrt{3}) + 3 (1)}{9} ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.10.3

Factor $3$ out of $3 (\sqrt{3}) + 3 (1)$ .

$\frac{3 (\sqrt{3} + 1)}{9} ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.10.4

Cancel the common factors.

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Step 4.1.2.10.4.1

Factor $3$ out of $9$ .

$\frac{3 (\sqrt{3} + 1)}{3 (3)} ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.10.4.2

Cancel the common factor.

$\frac{3 (\sqrt{3} + 1)}{3 \cdot 3} ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.10.4.3

Rewrite the expression.

$\frac{\sqrt{3} + 1}{3} ((\frac{6 + \sqrt{3}}{3}) - 3)$

Step 4.1.2.11

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

$\frac{\sqrt{3} + 1}{3} (\frac{6 + \sqrt{3}}{3} - 3 \cdot \frac{3}{3})$

Step 4.1.2.12

Combine fractions.

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Step 4.1.2.12.1

Combine $- 3$ and $\frac{3}{3}$ .

$\frac{\sqrt{3} + 1}{3} (\frac{6 + \sqrt{3}}{3} + \frac{- 3 \cdot 3}{3})$

Step 4.1.2.12.2

Combine the numerators over the common denominator.

$\frac{\sqrt{3} + 1}{3} \cdot \frac{6 + \sqrt{3} - 3 \cdot 3}{3}$

Step 4.1.2.13

Simplify the numerator.

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Step 4.1.2.13.1

Multiply $- 3$ by $3$ .

$\frac{\sqrt{3} + 1}{3} \cdot \frac{6 + \sqrt{3} - 9}{3}$

Step 4.1.2.13.2

Subtract $9$ from $6$ .

$\frac{\sqrt{3} + 1}{3} \cdot \frac{- 3 + \sqrt{3}}{3}$

Step 4.1.2.14

Multiply $\frac{\sqrt{3} + 1}{3} \cdot \frac{- 3 + \sqrt{3}}{3}$ .

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Step 4.1.2.14.1

Multiply $\frac{\sqrt{3} + 1}{3}$ by $\frac{- 3 + \sqrt{3}}{3}$ .

$\frac{(\sqrt{3} + 1) (- 3 + \sqrt{3})}{3 \cdot 3}$

Step 4.1.2.14.2

Multiply $3$ by $3$ .

$\frac{(\sqrt{3} + 1) (- 3 + \sqrt{3})}{9}$

Step 4.1.2.15

Simplify the numerator.

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Step 4.1.2.15.1

Expand $(\sqrt{3} + 1) (- 3 + \sqrt{3})$ using the FOIL Method.

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Step 4.1.2.15.1.1

Apply the distributive property.

$\frac{\sqrt{3} (- 3 + \sqrt{3}) + 1 (- 3 + \sqrt{3})}{9}$

Step 4.1.2.15.1.2

Apply the distributive property.

$\frac{\sqrt{3} \cdot - 3 + \sqrt{3} \sqrt{3} + 1 (- 3 + \sqrt{3})}{9}$

Step 4.1.2.15.1.3

Apply the distributive property.

$\frac{\sqrt{3} \cdot - 3 + \sqrt{3} \sqrt{3} + 1 \cdot - 3 + 1 \sqrt{3}}{9}$

Step 4.1.2.15.2

Simplify and combine like terms.

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Step 4.1.2.15.2.1

Simplify each term.

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Step 4.1.2.15.2.1.1

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

$\frac{- 3 \cdot \sqrt{3} + \sqrt{3} \sqrt{3} + 1 \cdot - 3 + 1 \sqrt{3}}{9}$

Step 4.1.2.15.2.1.2

Combine using the product rule for radicals.

$\frac{- 3 \sqrt{3} + \sqrt{3 \cdot 3} + 1 \cdot - 3 + 1 \sqrt{3}}{9}$

Step 4.1.2.15.2.1.3

Multiply $3$ by $3$ .

$\frac{- 3 \sqrt{3} + \sqrt{9} + 1 \cdot - 3 + 1 \sqrt{3}}{9}$

Step 4.1.2.15.2.1.4

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

$\frac{- 3 \sqrt{3} + \sqrt{3^{2}} + 1 \cdot - 3 + 1 \sqrt{3}}{9}$

Step 4.1.2.15.2.1.5

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

$\frac{- 3 \sqrt{3} + 3 + 1 \cdot - 3 + 1 \sqrt{3}}{9}$

Step 4.1.2.15.2.1.6

Multiply $- 3$ by $1$ .

$\frac{- 3 \sqrt{3} + 3 - 3 + 1 \sqrt{3}}{9}$

Step 4.1.2.15.2.1.7

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

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

Step 4.1.2.15.2.2

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

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

Step 4.1.2.15.2.3

Subtract $3$ from $3$ .

$\frac{0 - 2 \sqrt{3}}{9}$

Step 4.1.2.15.2.4

Subtract $2 \sqrt{3}$ from $0$ .

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

Step 4.1.2.16

Move the negative in front of the fraction.

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

Step 4.2

Evaluate at $x = \frac{6 - \sqrt{3}}{3}$ .

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Step 4.2.1

Substitute $\frac{6 - \sqrt{3}}{3}$ for $x$ .

$((\frac{6 - \sqrt{3}}{3}) - 1) ((\frac{6 - \sqrt{3}}{3}) - 2) ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2

Simplify.

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Step 4.2.2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$(\frac{6 - \sqrt{3}}{3} - 1 \cdot \frac{3}{3}) ((\frac{6 - \sqrt{3}}{3}) - 2) ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.2

Combine fractions.

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Step 4.2.2.2.1

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 4.2.2.2.2

Combine the numerators over the common denominator.

$\frac{6 - \sqrt{3} - 1 \cdot 3}{3} ((\frac{6 - \sqrt{3}}{3}) - 2) ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.3

Simplify the numerator.

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Step 4.2.2.3.1

Multiply $- 1$ by $3$ .

$\frac{6 - \sqrt{3} - 3}{3} ((\frac{6 - \sqrt{3}}{3}) - 2) ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.3.2

Subtract $3$ from $6$ .

$\frac{3 - \sqrt{3}}{3} ((\frac{6 - \sqrt{3}}{3}) - 2) ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.4

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

$\frac{3 - \sqrt{3}}{3} (\frac{6 - \sqrt{3}}{3} - 2 \cdot \frac{3}{3}) ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.5

Combine fractions.

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Step 4.2.2.5.1

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

$\frac{3 - \sqrt{3}}{3} (\frac{6 - \sqrt{3}}{3} + \frac{- 2 \cdot 3}{3}) ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.5.2

Combine the numerators over the common denominator.

$\frac{3 - \sqrt{3}}{3} \cdot \frac{6 - \sqrt{3} - 2 \cdot 3}{3} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.6

Simplify the numerator.

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Step 4.2.2.6.1

Multiply $- 2$ by $3$ .

$\frac{3 - \sqrt{3}}{3} \cdot \frac{6 - \sqrt{3} - 6}{3} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.6.2

Subtract $6$ from $6$ .

$\frac{3 - \sqrt{3}}{3} \cdot \frac{0 - \sqrt{3}}{3} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.6.3

Subtract $\sqrt{3}$ from $0$ .

$\frac{3 - \sqrt{3}}{3} \cdot \frac{- \sqrt{3}}{3} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.7

Move the negative in front of the fraction.

$\frac{3 - \sqrt{3}}{3} (- \frac{\sqrt{3}}{3}) ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.8

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

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Step 4.2.2.8.1

Multiply $\frac{3 - \sqrt{3}}{3}$ by $\frac{\sqrt{3}}{3}$ .

$- \frac{(3 - \sqrt{3}) \sqrt{3}}{3 \cdot 3} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.8.2

Multiply $3$ by $3$ .

$- \frac{(3 - \sqrt{3}) \sqrt{3}}{9} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.9

Apply the distributive property.

$- \frac{3 \sqrt{3} - \sqrt{3} \sqrt{3}}{9} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.10

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

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Step 4.2.2.10.1

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

$- \frac{3 \sqrt{3} - ({\sqrt{3}}^{1} \sqrt{3})}{9} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.10.2

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

$- \frac{3 \sqrt{3} - ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{9} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.10.3

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

$- \frac{3 \sqrt{3} - {\sqrt{3}}^{1 + 1}}{9} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.10.4

Add $1$ and $1$ .

$- \frac{3 \sqrt{3} - {\sqrt{3}}^{2}}{9} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.11

Simplify each term.

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Step 4.2.2.11.1

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

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Step 4.2.2.11.1.1

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

$- \frac{3 \sqrt{3} - {(3^{\frac{1}{2}})}^{2}}{9} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.11.1.2

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

$- \frac{3 \sqrt{3} - 3^{\frac{1}{2} \cdot 2}}{9} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.11.1.3

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

$- \frac{3 \sqrt{3} - 3^{\frac{2}{2}}}{9} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.11.1.4

Cancel the common factor of $2$ .

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Step 4.2.2.11.1.4.1

Cancel the common factor.

$- \frac{3 \sqrt{3} - 3^{\frac{2}{2}}}{9} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.11.1.4.2

Rewrite the expression.

$- \frac{3 \sqrt{3} - 3^{1}}{9} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.11.1.5

Evaluate the exponent.

$- \frac{3 \sqrt{3} - 1 \cdot 3}{9} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.11.2

Multiply $- 1$ by $3$ .

$- \frac{3 \sqrt{3} - 3}{9} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.12

Cancel the common factor of $3 \sqrt{3} - 3$ and $9$ .

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Step 4.2.2.12.1

Factor $3$ out of $3 \sqrt{3}$ .

$- \frac{3 (\sqrt{3}) - 3}{9} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.12.2

Factor $3$ out of $- 3$ .

$- \frac{3 (\sqrt{3}) + 3 (- 1)}{9} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.12.3

Factor $3$ out of $3 (\sqrt{3}) + 3 (- 1)$ .

$- \frac{3 (\sqrt{3} - 1)}{9} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.12.4

Cancel the common factors.

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Step 4.2.2.12.4.1

Factor $3$ out of $9$ .

$- \frac{3 (\sqrt{3} - 1)}{3 (3)} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.12.4.2

Cancel the common factor.

$- \frac{3 (\sqrt{3} - 1)}{3 \cdot 3} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.12.4.3

Rewrite the expression.

$- \frac{\sqrt{3} - 1}{3} ((\frac{6 - \sqrt{3}}{3}) - 3)$

Step 4.2.2.13

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

$- \frac{\sqrt{3} - 1}{3} (\frac{6 - \sqrt{3}}{3} - 3 \cdot \frac{3}{3})$

Step 4.2.2.14

Combine fractions.

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Step 4.2.2.14.1

Combine $- 3$ and $\frac{3}{3}$ .

$- \frac{\sqrt{3} - 1}{3} (\frac{6 - \sqrt{3}}{3} + \frac{- 3 \cdot 3}{3})$

Step 4.2.2.14.2

Combine the numerators over the common denominator.

$- \frac{\sqrt{3} - 1}{3} \cdot \frac{6 - \sqrt{3} - 3 \cdot 3}{3}$

Step 4.2.2.15

Simplify the numerator.

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Step 4.2.2.15.1

Multiply $- 3$ by $3$ .

$- \frac{\sqrt{3} - 1}{3} \cdot \frac{6 - \sqrt{3} - 9}{3}$

Step 4.2.2.15.2

Subtract $9$ from $6$ .

$- \frac{\sqrt{3} - 1}{3} \cdot \frac{- 3 - \sqrt{3}}{3}$

Step 4.2.2.16

Multiply $- \frac{\sqrt{3} - 1}{3} \cdot \frac{- 3 - \sqrt{3}}{3}$ .

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Step 4.2.2.16.1

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

$- \frac{(- 3 - \sqrt{3}) (\sqrt{3} - 1)}{3 \cdot 3}$

Step 4.2.2.16.2

Multiply $3$ by $3$ .

$- \frac{(- 3 - \sqrt{3}) (\sqrt{3} - 1)}{9}$

Step 4.2.2.17

Simplify the numerator.

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Step 4.2.2.17.1

Expand $(- 3 - \sqrt{3}) (\sqrt{3} - 1)$ using the FOIL Method.

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Step 4.2.2.17.1.1

Apply the distributive property.

$- \frac{- 3 (\sqrt{3} - 1) - \sqrt{3} (\sqrt{3} - 1)}{9}$

Step 4.2.2.17.1.2

Apply the distributive property.

$- \frac{- 3 \sqrt{3} - 3 \cdot - 1 - \sqrt{3} (\sqrt{3} - 1)}{9}$

Step 4.2.2.17.1.3

Apply the distributive property.

$- \frac{- 3 \sqrt{3} - 3 \cdot - 1 - \sqrt{3} \sqrt{3} - \sqrt{3} \cdot - 1}{9}$

Step 4.2.2.17.2

Simplify and combine like terms.

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Step 4.2.2.17.2.1

Simplify each term.

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Step 4.2.2.17.2.1.1

Multiply $- 3$ by $- 1$ .

$- \frac{- 3 \sqrt{3} + 3 - \sqrt{3} \sqrt{3} - \sqrt{3} \cdot - 1}{9}$

Step 4.2.2.17.2.1.2

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

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Step 4.2.2.17.2.1.2.1

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

$- \frac{- 3 \sqrt{3} + 3 - ({\sqrt{3}}^{1} \sqrt{3}) - \sqrt{3} \cdot - 1}{9}$

Step 4.2.2.17.2.1.2.2

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

$- \frac{- 3 \sqrt{3} + 3 - ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) - \sqrt{3} \cdot - 1}{9}$

Step 4.2.2.17.2.1.2.3

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

$- \frac{- 3 \sqrt{3} + 3 - {\sqrt{3}}^{1 + 1} - \sqrt{3} \cdot - 1}{9}$

Step 4.2.2.17.2.1.2.4

Add $1$ and $1$ .

$- \frac{- 3 \sqrt{3} + 3 - {\sqrt{3}}^{2} - \sqrt{3} \cdot - 1}{9}$

Step 4.2.2.17.2.1.3

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

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Step 4.2.2.17.2.1.3.1

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

$- \frac{- 3 \sqrt{3} + 3 - {(3^{\frac{1}{2}})}^{2} - \sqrt{3} \cdot - 1}{9}$

Step 4.2.2.17.2.1.3.2

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

$- \frac{- 3 \sqrt{3} + 3 - 3^{\frac{1}{2} \cdot 2} - \sqrt{3} \cdot - 1}{9}$

Step 4.2.2.17.2.1.3.3

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

$- \frac{- 3 \sqrt{3} + 3 - 3^{\frac{2}{2}} - \sqrt{3} \cdot - 1}{9}$

Step 4.2.2.17.2.1.3.4

Cancel the common factor of $2$ .

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Step 4.2.2.17.2.1.3.4.1

Cancel the common factor.

$- \frac{- 3 \sqrt{3} + 3 - 3^{\frac{2}{2}} - \sqrt{3} \cdot - 1}{9}$

Step 4.2.2.17.2.1.3.4.2

Rewrite the expression.

$- \frac{- 3 \sqrt{3} + 3 - 3^{1} - \sqrt{3} \cdot - 1}{9}$

Step 4.2.2.17.2.1.3.5

Evaluate the exponent.

$- \frac{- 3 \sqrt{3} + 3 - 1 \cdot 3 - \sqrt{3} \cdot - 1}{9}$

Step 4.2.2.17.2.1.4

Multiply $- 1$ by $3$ .

$- \frac{- 3 \sqrt{3} + 3 - 3 - \sqrt{3} \cdot - 1}{9}$

Step 4.2.2.17.2.1.5

Multiply $- \sqrt{3} \cdot - 1$ .

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Step 4.2.2.17.2.1.5.1

Multiply $- 1$ by $- 1$ .

$- \frac{- 3 \sqrt{3} + 3 - 3 + 1 \sqrt{3}}{9}$

Step 4.2.2.17.2.1.5.2

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

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

Step 4.2.2.17.2.2

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

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

Step 4.2.2.17.2.3

Subtract $3$ from $3$ .

$- \frac{0 - 2 \sqrt{3}}{9}$

Step 4.2.2.17.2.4

Subtract $2 \sqrt{3}$ from $0$ .

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

Step 4.2.2.18

Move the negative in front of the fraction.

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

Step 4.2.2.19

Multiply $- - \frac{2 \sqrt{3}}{9}$ .

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Step 4.2.2.19.1

Multiply $- 1$ by $- 1$ .

$1 \frac{2 \sqrt{3}}{9}$

Step 4.2.2.19.2

Multiply $\frac{2 \sqrt{3}}{9}$ by $1$ .

$\frac{2 \sqrt{3}}{9}$

Step 4.3

List all of the points.

$(\frac{6 + \sqrt{3}}{3}, - \frac{2 \sqrt{3}}{9}), (\frac{6 - \sqrt{3}}{3}, \frac{2 \sqrt{3}}{9})$

Step 5