Calculus Examples

$f (x) = \frac{x - 1}{x^{2} - 5 x + 6}$

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 Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x - 1$ and $g (x) = x^{2} - 5 x + 6$ .

$\frac{(x^{2} - 5 x + 6) \frac{d}{d x} [x - 1] - (x - 1) \frac{d}{d x} [x^{2} - 5 x + 6]}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

$\frac{(x^{2} - 5 x + 6) (\frac{d}{d x} [x] + \frac{d}{d x} [- 1]) - (x - 1) \frac{d}{d x} [x^{2} - 5 x + 6]}{{(x^{2} - 5 x + 6)}^{2}}$

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 $n = 1$ .

$\frac{(x^{2} - 5 x + 6) (1 + \frac{d}{d x} [- 1]) - (x - 1) \frac{d}{d x} [x^{2} - 5 x + 6]}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.2.3

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

$\frac{(x^{2} - 5 x + 6) (1 + 0) - (x - 1) \frac{d}{d x} [x^{2} - 5 x + 6]}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.2.4

Simplify the expression.

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Step 1.1.2.4.1

Add $1$ and $0$ .

$\frac{(x^{2} - 5 x + 6) \cdot 1 - (x - 1) \frac{d}{d x} [x^{2} - 5 x + 6]}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.2.4.2

Multiply $x^{2} - 5 x + 6$ by $1$ .

$\frac{x^{2} - 5 x + 6 - (x - 1) \frac{d}{d x} [x^{2} - 5 x + 6]}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.2.5

By the Sum Rule, the derivative of $x^{2} - 5 x + 6$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [6]$ .

$\frac{x^{2} - 5 x + 6 - (x - 1) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [6])}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.2.6

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

$\frac{x^{2} - 5 x + 6 - (x - 1) (2 x + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [6])}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.2.7

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$\frac{x^{2} - 5 x + 6 - (x - 1) (2 x - 5 \frac{d}{d x} [x] + \frac{d}{d x} [6])}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.2.8

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

$\frac{x^{2} - 5 x + 6 - (x - 1) (2 x - 5 \cdot 1 + \frac{d}{d x} [6])}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.2.9

Multiply $- 5$ by $1$ .

$\frac{x^{2} - 5 x + 6 - (x - 1) (2 x - 5 + \frac{d}{d x} [6])}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.2.10

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$\frac{x^{2} - 5 x + 6 - (x - 1) (2 x - 5 + 0)}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.2.11

Add $2 x - 5$ and $0$ .

$\frac{x^{2} - 5 x + 6 - (x - 1) (2 x - 5)}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$\frac{x^{2} - 5 x + 6 + (- x - - 1) (2 x - 5)}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.3.2

Simplify the numerator.

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Step 1.1.3.2.1

Simplify each term.

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Step 1.1.3.2.1.1

Multiply $- 1$ by $- 1$ .

$\frac{x^{2} - 5 x + 6 + (- x + 1) (2 x - 5)}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.3.2.1.2

Expand $(- x + 1) (2 x - 5)$ using the FOIL Method.

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Step 1.1.3.2.1.2.1

Apply the distributive property.

$\frac{x^{2} - 5 x + 6 - x (2 x - 5) + 1 (2 x - 5)}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.3.2.1.2.2

Apply the distributive property.

$\frac{x^{2} - 5 x + 6 - x (2 x) - x \cdot - 5 + 1 (2 x - 5)}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.3.2.1.2.3

Apply the distributive property.

$\frac{x^{2} - 5 x + 6 - x (2 x) - x \cdot - 5 + 1 (2 x) + 1 \cdot - 5}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.3.2.1.3

Simplify and combine like terms.

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Step 1.1.3.2.1.3.1

Simplify each term.

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Step 1.1.3.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{x^{2} - 5 x + 6 - 1 \cdot 2 x \cdot x - x \cdot - 5 + 1 (2 x) + 1 \cdot - 5}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.3.2.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.3.2.1.3.1.2.1

Move $x$ .

$\frac{x^{2} - 5 x + 6 - 1 \cdot 2 (x \cdot x) - x \cdot - 5 + 1 (2 x) + 1 \cdot - 5}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.3.2.1.3.1.2.2

Multiply $x$ by $x$ .

$\frac{x^{2} - 5 x + 6 - 1 \cdot 2 x^{2} - x \cdot - 5 + 1 (2 x) + 1 \cdot - 5}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.3.2.1.3.1.3

Multiply $- 1$ by $2$ .

$\frac{x^{2} - 5 x + 6 - 2 x^{2} - x \cdot - 5 + 1 (2 x) + 1 \cdot - 5}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.3.2.1.3.1.4

Multiply $- 5$ by $- 1$ .

$\frac{x^{2} - 5 x + 6 - 2 x^{2} + 5 x + 1 (2 x) + 1 \cdot - 5}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.3.2.1.3.1.5

Multiply $2 x$ by $1$ .

$\frac{x^{2} - 5 x + 6 - 2 x^{2} + 5 x + 2 x + 1 \cdot - 5}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.3.2.1.3.1.6

Multiply $- 5$ by $1$ .

$\frac{x^{2} - 5 x + 6 - 2 x^{2} + 5 x + 2 x - 5}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.3.2.1.3.2

Add $5 x$ and $2 x$ .

$\frac{x^{2} - 5 x + 6 - 2 x^{2} + 7 x - 5}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.3.2.2

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

$\frac{- x^{2} - 5 x + 6 + 7 x - 5}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.3.2.3

Add $- 5 x$ and $7 x$ .

$\frac{- x^{2} + 2 x + 6 - 5}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.3.2.4

Subtract $5$ from $6$ .

$\frac{- x^{2} + 2 x + 1}{{(x^{2} - 5 x + 6)}^{2}}$

Step 1.1.3.3

Simplify the denominator.

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Step 1.1.3.3.1

Factor $x^{2} - 5 x + 6$ using the AC method.

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Step 1.1.3.3.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $6$ and whose sum is $- 5$ .

$- 3, - 2$

Step 1.1.3.3.1.2

Write the factored form using these integers.

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

Step 1.1.3.3.2

Apply the product rule to $(x - 3) (x - 2)$ .

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

Step 1.1.3.4

Factor $- 1$ out of $- x^{2}$ .

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

Step 1.1.3.5

Factor $- 1$ out of $2 x$ .

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

Step 1.1.3.6

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

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

Step 1.1.3.7

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

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

Step 1.1.3.8

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

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

Step 1.1.3.9

Rewrite $- (x^{2} - 2 x - 1)$ as $- 1 (x^{2} - 2 x - 1)$ .

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

Step 1.1.3.10

Move the negative in front of the fraction.

$f' (x) = - \frac{x^{2} - 2 x - 1}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{x^{2} - 2 x - 1}{{(x - 3)}^{2} {(x - 2)}^{2}}$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{x^{2} - 2 x - 1}{{(x - 3)}^{2} {(x - 2)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{x^{2} - 2 x - 1}{{(x - 3)}^{2} {(x - 2)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

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

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Use the quadratic formula to find the solutions.

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

Step 2.3.2

Substitute the values $a = 1$ , $b = - 2$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

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

Step 2.3.3

Simplify.

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Step 2.3.3.1

Simplify the numerator.

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Step 2.3.3.1.1

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

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

Step 2.3.3.1.2

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

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Step 2.3.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.3.3.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 2.3.3.1.3

Add $4$ and $4$ .

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

Step 2.3.3.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.3.3.1.4.1

Factor $4$ out of $8$ .

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

Step 2.3.3.1.4.2

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

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

Step 2.3.3.1.5

Pull terms out from under the radical.

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

Step 2.3.3.2

Multiply $2$ by $1$ .

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

Step 2.3.3.3

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

$x = 1 \pm \sqrt{2}$

Step 2.3.4

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

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Step 2.3.4.1

Simplify the numerator.

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Step 2.3.4.1.1

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

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

Step 2.3.4.1.2

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

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Step 2.3.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.3.4.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 2.3.4.1.3

Add $4$ and $4$ .

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

Step 2.3.4.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.3.4.1.4.1

Factor $4$ out of $8$ .

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

Step 2.3.4.1.4.2

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

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

Step 2.3.4.1.5

Pull terms out from under the radical.

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

Step 2.3.4.2

Multiply $2$ by $1$ .

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

Step 2.3.4.3

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

$x = 1 \pm \sqrt{2}$

Step 2.3.4.4

Change the $\pm$ to $+$ .

$x = 1 + \sqrt{2}$

Step 2.3.5

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

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Step 2.3.5.1

Simplify the numerator.

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Step 2.3.5.1.1

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

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

Step 2.3.5.1.2

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

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Step 2.3.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.3.5.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 2.3.5.1.3

Add $4$ and $4$ .

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

Step 2.3.5.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.3.5.1.4.1

Factor $4$ out of $8$ .

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

Step 2.3.5.1.4.2

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

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

Step 2.3.5.1.5

Pull terms out from under the radical.

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

Step 2.3.5.2

Multiply $2$ by $1$ .

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

Step 2.3.5.3

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

$x = 1 \pm \sqrt{2}$

Step 2.3.5.4

Change the $\pm$ to $-$ .

$x = 1 - \sqrt{2}$

Step 2.3.6

The final answer is the combination of both solutions.

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

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{x^{2} - 2 x - 1}{{(x - 3)}^{2} {(x - 2)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x - 3)}^{2} {(x - 2)}^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

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

${(x - 3)}^{2} = 0$

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

Step 3.2.2

Set ${(x - 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.2.2.1

Set ${(x - 3)}^{2}$ equal to $0$ .

${(x - 3)}^{2} = 0$

Step 3.2.2.2

Solve ${(x - 3)}^{2} = 0$ for $x$ .

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Step 3.2.2.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 3.2.2.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 3.2.3

Set ${(x - 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.2.3.1

Set ${(x - 2)}^{2}$ equal to $0$ .

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

Step 3.2.3.2

Solve ${(x - 2)}^{2} = 0$ for $x$ .

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Step 3.2.3.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 3.2.3.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.2.4

The final solution is all the values that make ${(x - 3)}^{2} {(x - 2)}^{2} = 0$ true.

$x = 3, 2$

Step 3.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = 2, x = 3$

Step 4

Evaluate $\frac{x - 1}{x^{2} - 5 x + 6}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the numerator.

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Step 4.1.2.1.1

Subtract $1$ from $1$ .

$\frac{0 + \sqrt{2}}{{(1 + \sqrt{2})}^{2} - 5 (1 + \sqrt{2}) + 6}$

Step 4.1.2.1.2

Add $0$ and $\sqrt{2}$ .

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

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

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

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

Step 4.1.2.2.2

Expand $(1 + \sqrt{2}) (1 + \sqrt{2})$ using the FOIL Method.

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Step 4.1.2.2.2.1

Apply the distributive property.

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

Step 4.1.2.2.2.2

Apply the distributive property.

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

Step 4.1.2.2.2.3

Apply the distributive property.

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

Step 4.1.2.2.3

Simplify and combine like terms.

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Step 4.1.2.2.3.1

Simplify each term.

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Step 4.1.2.2.3.1.1

Multiply $1$ by $1$ .

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

Step 4.1.2.2.3.1.2

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

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

Step 4.1.2.2.3.1.3

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

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

Step 4.1.2.2.3.1.4

Combine using the product rule for radicals.

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

Step 4.1.2.2.3.1.5

Multiply $2$ by $2$ .

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

Step 4.1.2.2.3.1.6

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

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

Step 4.1.2.2.3.1.7

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

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

Step 4.1.2.2.3.2

Add $1$ and $2$ .

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

Step 4.1.2.2.3.3

Add $\sqrt{2}$ and $\sqrt{2}$ .

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

Step 4.1.2.2.4

Apply the distributive property.

$\frac{\sqrt{2}}{3 + 2 \sqrt{2} - 5 \cdot 1 - 5 \sqrt{2} + 6}$

Step 4.1.2.2.5

Multiply $- 5$ by $1$ .

$\frac{\sqrt{2}}{3 + 2 \sqrt{2} - 5 - 5 \sqrt{2} + 6}$

Step 4.1.2.2.6

Subtract $5$ from $3$ .

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

Step 4.1.2.2.7

Add $- 2$ and $6$ .

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

Step 4.1.2.2.8

Subtract $5 \sqrt{2}$ from $2 \sqrt{2}$ .

$\frac{\sqrt{2}}{4 - 3 \sqrt{2}}$

Step 4.1.2.3

Multiply $\frac{\sqrt{2}}{4 - 3 \sqrt{2}}$ by $\frac{4 + 3 \sqrt{2}}{4 + 3 \sqrt{2}}$ .

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

Step 4.1.2.4

Multiply $\frac{\sqrt{2}}{4 - 3 \sqrt{2}}$ by $\frac{4 + 3 \sqrt{2}}{4 + 3 \sqrt{2}}$ .

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

Step 4.1.2.5

Expand the denominator using the FOIL method.

$\frac{\sqrt{2} (4 + 3 \sqrt{2})}{16 + 12 \sqrt{2} - 12 \sqrt{2} - 9 {\sqrt{2}}^{2}}$

Step 4.1.2.6

Simplify.

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

Step 4.1.2.7

Apply the distributive property.

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

Step 4.1.2.8

Move $4$ to the left of $\sqrt{2}$ .

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

Step 4.1.2.9

Multiply $\sqrt{2} (3 \sqrt{2})$ .

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Step 4.1.2.9.1

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.1.2.9.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.1.2.9.3

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

$\frac{4 \cdot \sqrt{2} + 3 {\sqrt{2}}^{1 + 1}}{- 2}$

Step 4.1.2.9.4

Add $1$ and $1$ .

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

Step 4.1.2.10

Simplify each term.

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Step 4.1.2.10.1

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

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Step 4.1.2.10.1.1

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

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

Step 4.1.2.10.1.2

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

$\frac{4 \sqrt{2} + 3 \cdot 2^{\frac{1}{2} \cdot 2}}{- 2}$

Step 4.1.2.10.1.3

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

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

Step 4.1.2.10.1.4

Cancel the common factor of $2$ .

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Step 4.1.2.10.1.4.1

Cancel the common factor.

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

Step 4.1.2.10.1.4.2

Rewrite the expression.

$\frac{4 \sqrt{2} + 3 \cdot 2^{1}}{- 2}$

Step 4.1.2.10.1.5

Evaluate the exponent.

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

Step 4.1.2.10.2

Multiply $3$ by $2$ .

$\frac{4 \sqrt{2} + 6}{- 2}$

Step 4.1.2.11

Simplify terms.

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Step 4.1.2.11.1

Cancel the common factor of $4 \sqrt{2} + 6$ and $- 2$ .

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Step 4.1.2.11.1.1

Factor $2$ out of $4 \sqrt{2}$ .

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

Step 4.1.2.11.1.2

Factor $2$ out of $6$ .

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

Step 4.1.2.11.1.3

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

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

Step 4.1.2.11.1.4

Move the negative one from the denominator of $\frac{2 \sqrt{2} + 3}{- 1}$ .

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

Step 4.1.2.11.2

Rewrite $- 1 \cdot (2 \sqrt{2} + 3)$ as $- (2 \sqrt{2} + 3)$ .

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

Step 4.1.2.11.3

Apply the distributive property.

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

Step 4.1.2.11.4

Multiply.

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Step 4.1.2.11.4.1

Multiply $2$ by $- 1$ .

$- 2 \sqrt{2} - 1 \cdot 3$

Step 4.1.2.11.4.2

Multiply $- 1$ by $3$ .

$- 2 \sqrt{2} - 3$

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify the numerator.

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Step 4.2.2.1.1

Subtract $1$ from $1$ .

$\frac{0 - \sqrt{2}}{{(1 - \sqrt{2})}^{2} - 5 (1 - \sqrt{2}) + 6}$

Step 4.2.2.1.2

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

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

Step 4.2.2.2

Simplify the denominator.

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Step 4.2.2.2.1

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

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

Step 4.2.2.2.2

Expand $(1 - \sqrt{2}) (1 - \sqrt{2})$ using the FOIL Method.

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Step 4.2.2.2.2.1

Apply the distributive property.

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

Step 4.2.2.2.2.2

Apply the distributive property.

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

Step 4.2.2.2.2.3

Apply the distributive property.

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

Step 4.2.2.2.3

Simplify and combine like terms.

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Step 4.2.2.2.3.1

Simplify each term.

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Step 4.2.2.2.3.1.1

Multiply $1$ by $1$ .

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

Step 4.2.2.2.3.1.2

Multiply $- \sqrt{2}$ by $1$ .

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

Step 4.2.2.2.3.1.3

Multiply $- 1$ by $1$ .

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

Step 4.2.2.2.3.1.4

Multiply $- \sqrt{2} (- \sqrt{2})$ .

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Step 4.2.2.2.3.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2.2.2.3.1.4.2

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

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

Step 4.2.2.2.3.1.4.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.2.2.2.3.1.4.4

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.2.2.2.3.1.4.5

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

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

Step 4.2.2.2.3.1.4.6

Add $1$ and $1$ .

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

Step 4.2.2.2.3.1.5

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

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Step 4.2.2.2.3.1.5.1

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

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

Step 4.2.2.2.3.1.5.2

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

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

Step 4.2.2.2.3.1.5.3

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

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

Step 4.2.2.2.3.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.2.2.3.1.5.4.1

Cancel the common factor.

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

Step 4.2.2.2.3.1.5.4.2

Rewrite the expression.

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

Step 4.2.2.2.3.1.5.5

Evaluate the exponent.

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

Step 4.2.2.2.3.2

Add $1$ and $2$ .

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

Step 4.2.2.2.3.3

Subtract $\sqrt{2}$ from $- \sqrt{2}$ .

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

Step 4.2.2.2.4

Apply the distributive property.

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

Step 4.2.2.2.5

Multiply $- 5$ by $1$ .

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

Step 4.2.2.2.6

Multiply $- 1$ by $- 5$ .

$\frac{- \sqrt{2}}{3 - 2 \sqrt{2} - 5 + 5 \sqrt{2} + 6}$

Step 4.2.2.2.7

Subtract $5$ from $3$ .

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

Step 4.2.2.2.8

Add $- 2$ and $6$ .

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

Step 4.2.2.2.9

Add $- 2 \sqrt{2}$ and $5 \sqrt{2}$ .

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

Step 4.2.2.3

Move the negative in front of the fraction.

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

Step 4.2.2.4

Multiply $\frac{\sqrt{2}}{4 + 3 \sqrt{2}}$ by $\frac{4 - 3 \sqrt{2}}{4 - 3 \sqrt{2}}$ .

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

Step 4.2.2.5

Multiply $\frac{\sqrt{2}}{4 + 3 \sqrt{2}}$ by $\frac{4 - 3 \sqrt{2}}{4 - 3 \sqrt{2}}$ .

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

Step 4.2.2.6

Expand the denominator using the FOIL method.

$- \frac{\sqrt{2} (4 - 3 \sqrt{2})}{16 - 12 \sqrt{2} + 12 \sqrt{2} - 9 {\sqrt{2}}^{2}}$

Step 4.2.2.7

Simplify.

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

Step 4.2.2.8

Apply the distributive property.

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

Step 4.2.2.9

Move $4$ to the left of $\sqrt{2}$ .

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

Step 4.2.2.10

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

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Step 4.2.2.10.1

Raise $\sqrt{2}$ to the power of $1$ .

$- \frac{4 \cdot \sqrt{2} - 3 ({\sqrt{2}}^{1} \sqrt{2})}{- 2}$

Step 4.2.2.10.2

Raise $\sqrt{2}$ to the power of $1$ .

$- \frac{4 \cdot \sqrt{2} - 3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{- 2}$

Step 4.2.2.10.3

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

$- \frac{4 \cdot \sqrt{2} - 3 {\sqrt{2}}^{1 + 1}}{- 2}$

Step 4.2.2.10.4

Add $1$ and $1$ .

$- \frac{4 \cdot \sqrt{2} - 3 {\sqrt{2}}^{2}}{- 2}$

Step 4.2.2.11

Simplify each term.

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Step 4.2.2.11.1

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

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Step 4.2.2.11.1.1

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

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

Step 4.2.2.11.1.2

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

$- \frac{4 \sqrt{2} - 3 \cdot 2^{\frac{1}{2} \cdot 2}}{- 2}$

Step 4.2.2.11.1.3

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

$- \frac{4 \sqrt{2} - 3 \cdot 2^{\frac{2}{2}}}{- 2}$

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.

$- \frac{4 \sqrt{2} - 3 \cdot 2^{\frac{2}{2}}}{- 2}$

Step 4.2.2.11.1.4.2

Rewrite the expression.

$- \frac{4 \sqrt{2} - 3 \cdot 2^{1}}{- 2}$

Step 4.2.2.11.1.5

Evaluate the exponent.

$- \frac{4 \sqrt{2} - 3 \cdot 2}{- 2}$

Step 4.2.2.11.2

Multiply $- 3$ by $2$ .

$- \frac{4 \sqrt{2} - 6}{- 2}$

Step 4.2.2.12

Simplify terms.

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Step 4.2.2.12.1

Cancel the common factor of $4 \sqrt{2} - 6$ and $- 2$ .

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Step 4.2.2.12.1.1

Factor $2$ out of $4 \sqrt{2}$ .

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

Step 4.2.2.12.1.2

Factor $2$ out of $- 6$ .

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

Step 4.2.2.12.1.3

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

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

Step 4.2.2.12.1.4

Move the negative one from the denominator of $\frac{2 \sqrt{2} - 3}{- 1}$ .

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

Step 4.2.2.12.2

Rewrite $- 1 \cdot (2 \sqrt{2} - 3)$ as $- (2 \sqrt{2} - 3)$ .

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

Step 4.2.2.12.3

Apply the distributive property.

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

Step 4.2.2.12.4

Multiply.

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Step 4.2.2.12.4.1

Multiply $2$ by $- 1$ .

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

Step 4.2.2.12.4.2

Multiply $- 1$ by $- 3$ .

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

Step 4.2.2.12.5

Apply the distributive property.

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

Step 4.2.2.12.6

Multiply.

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Step 4.2.2.12.6.1

Multiply $- 2$ by $- 1$ .

$2 \sqrt{2} - 1 \cdot 3$

Step 4.2.2.12.6.2

Multiply $- 1$ by $3$ .

$2 \sqrt{2} - 3$

Step 4.3

Evaluate at $x = 2$ .

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Step 4.3.1

Substitute $2$ for $x$ .

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

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

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

$\frac{(2) - 1}{4 - 5 \cdot 2 + 6}$

Step 4.3.2.1.2

Multiply $- 5$ by $2$ .

$\frac{(2) - 1}{4 - 10 + 6}$

Step 4.3.2.2

Simplify by adding and subtracting.

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Step 4.3.2.2.1

Subtract $10$ from $4$ .

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

Step 4.3.2.2.2

Add $- 6$ and $6$ .

$\frac{(2) - 1}{0}$

Step 4.3.2.2.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{(2) - 1}{0}$

Step 4.3.2.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.4

Evaluate at $x = 3$ .

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Step 4.4.1

Substitute $3$ for $x$ .

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

Step 4.4.2

Simplify.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

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

$\frac{(3) - 1}{9 - 5 \cdot 3 + 6}$

Step 4.4.2.1.2

Multiply $- 5$ by $3$ .

$\frac{(3) - 1}{9 - 15 + 6}$

Step 4.4.2.2

Simplify by adding and subtracting.

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Step 4.4.2.2.1

Subtract $15$ from $9$ .

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

Step 4.4.2.2.2

Add $- 6$ and $6$ .

$\frac{(3) - 1}{0}$

Step 4.4.2.2.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{(3) - 1}{0}$

Step 4.4.2.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.5

List all of the points.

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

Step 5