Calculus Examples

$y = \frac{x^{2} - 6 x + 37}{x - 6}$

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 $f (x) = x^{2} - 6 x + 37$ and $g (x) = x - 6$ .

$\frac{(x - 6) \frac{d}{d x} [x^{2} - 6 x + 37] - (x^{2} - 6 x + 37) \frac{d}{d x} [x - 6]}{{(x - 6)}^{2}}$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $x^{2} - 6 x + 37$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [37]$ .

$\frac{(x - 6) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [37]) - (x^{2} - 6 x + 37) \frac{d}{d x} [x - 6]}{{(x - 6)}^{2}}$

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 $n = 2$ .

$\frac{(x - 6) (2 x + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [37]) - (x^{2} - 6 x + 37) \frac{d}{d x} [x - 6]}{{(x - 6)}^{2}}$

Step 1.1.2.3

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

$\frac{(x - 6) (2 x - 6 \frac{d}{d x} [x] + \frac{d}{d x} [37]) - (x^{2} - 6 x + 37) \frac{d}{d x} [x - 6]}{{(x - 6)}^{2}}$

Step 1.1.2.4

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

$\frac{(x - 6) (2 x - 6 \cdot 1 + \frac{d}{d x} [37]) - (x^{2} - 6 x + 37) \frac{d}{d x} [x - 6]}{{(x - 6)}^{2}}$

Step 1.1.2.5

Multiply $- 6$ by $1$ .

$\frac{(x - 6) (2 x - 6 + \frac{d}{d x} [37]) - (x^{2} - 6 x + 37) \frac{d}{d x} [x - 6]}{{(x - 6)}^{2}}$

Step 1.1.2.6

Since $37$ is constant with respect to $x$ , the derivative of $37$ with respect to $x$ is $0$ .

$\frac{(x - 6) (2 x - 6 + 0) - (x^{2} - 6 x + 37) \frac{d}{d x} [x - 6]}{{(x - 6)}^{2}}$

Step 1.1.2.7

Add $2 x - 6$ and $0$ .

$\frac{(x - 6) (2 x - 6) - (x^{2} - 6 x + 37) \frac{d}{d x} [x - 6]}{{(x - 6)}^{2}}$

Step 1.1.2.8

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

$\frac{(x - 6) (2 x - 6) - (x^{2} - 6 x + 37) (\frac{d}{d x} [x] + \frac{d}{d x} [- 6])}{{(x - 6)}^{2}}$

Step 1.1.2.9

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

$\frac{(x - 6) (2 x - 6) - (x^{2} - 6 x + 37) (1 + \frac{d}{d x} [- 6])}{{(x - 6)}^{2}}$

Step 1.1.2.10

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

$\frac{(x - 6) (2 x - 6) - (x^{2} - 6 x + 37) (1 + 0)}{{(x - 6)}^{2}}$

Step 1.1.2.11

Simplify the expression.

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Step 1.1.2.11.1

Add $1$ and $0$ .

$\frac{(x - 6) (2 x - 6) - (x^{2} - 6 x + 37) \cdot 1}{{(x - 6)}^{2}}$

Step 1.1.2.11.2

Multiply $- 1$ by $1$ .

$\frac{(x - 6) (2 x - 6) - (x^{2} - 6 x + 37)}{{(x - 6)}^{2}}$

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$\frac{(x - 6) (2 x - 6) - x^{2} - (- 6 x) - 1 \cdot 37}{{(x - 6)}^{2}}$

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

Expand $(x - 6) (2 x - 6)$ using the FOIL Method.

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Step 1.1.3.2.1.1.1

Apply the distributive property.

$\frac{x (2 x - 6) - 6 (2 x - 6) - x^{2} - (- 6 x) - 1 \cdot 37}{{(x - 6)}^{2}}$

Step 1.1.3.2.1.1.2

Apply the distributive property.

$\frac{x (2 x) + x \cdot - 6 - 6 (2 x - 6) - x^{2} - (- 6 x) - 1 \cdot 37}{{(x - 6)}^{2}}$

Step 1.1.3.2.1.1.3

Apply the distributive property.

$\frac{x (2 x) + x \cdot - 6 - 6 (2 x) - 6 \cdot - 6 - x^{2} - (- 6 x) - 1 \cdot 37}{{(x - 6)}^{2}}$

Step 1.1.3.2.1.2

Simplify and combine like terms.

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Step 1.1.3.2.1.2.1

Simplify each term.

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Step 1.1.3.2.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x \cdot x + x \cdot - 6 - 6 (2 x) - 6 \cdot - 6 - x^{2} - (- 6 x) - 1 \cdot 37}{{(x - 6)}^{2}}$

Step 1.1.3.2.1.2.1.2

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

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Step 1.1.3.2.1.2.1.2.1

Move $x$ .

$\frac{2 (x \cdot x) + x \cdot - 6 - 6 (2 x) - 6 \cdot - 6 - x^{2} - (- 6 x) - 1 \cdot 37}{{(x - 6)}^{2}}$

Step 1.1.3.2.1.2.1.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + x \cdot - 6 - 6 (2 x) - 6 \cdot - 6 - x^{2} - (- 6 x) - 1 \cdot 37}{{(x - 6)}^{2}}$

Step 1.1.3.2.1.2.1.3

Move $- 6$ to the left of $x$ .

$\frac{2 x^{2} - 6 \cdot x - 6 (2 x) - 6 \cdot - 6 - x^{2} - (- 6 x) - 1 \cdot 37}{{(x - 6)}^{2}}$

Step 1.1.3.2.1.2.1.4

Multiply $2$ by $- 6$ .

$\frac{2 x^{2} - 6 x - 12 x - 6 \cdot - 6 - x^{2} - (- 6 x) - 1 \cdot 37}{{(x - 6)}^{2}}$

Step 1.1.3.2.1.2.1.5

Multiply $- 6$ by $- 6$ .

$\frac{2 x^{2} - 6 x - 12 x + 36 - x^{2} - (- 6 x) - 1 \cdot 37}{{(x - 6)}^{2}}$

Step 1.1.3.2.1.2.2

Subtract $12 x$ from $- 6 x$ .

$\frac{2 x^{2} - 18 x + 36 - x^{2} - (- 6 x) - 1 \cdot 37}{{(x - 6)}^{2}}$

Step 1.1.3.2.1.3

Multiply $- 6$ by $- 1$ .

$\frac{2 x^{2} - 18 x + 36 - x^{2} + 6 x - 1 \cdot 37}{{(x - 6)}^{2}}$

Step 1.1.3.2.1.4

Multiply $- 1$ by $37$ .

$\frac{2 x^{2} - 18 x + 36 - x^{2} + 6 x - 37}{{(x - 6)}^{2}}$

Step 1.1.3.2.2

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

$\frac{x^{2} - 18 x + 36 + 6 x - 37}{{(x - 6)}^{2}}$

Step 1.1.3.2.3

Add $- 18 x$ and $6 x$ .

$\frac{x^{2} - 12 x + 36 - 37}{{(x - 6)}^{2}}$

Step 1.1.3.2.4

Subtract $37$ from $36$ .

$f' (x) = \frac{x^{2} - 12 x - 1}{{(x - 6)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x^{2} - 12 x - 1}{{(x - 6)}^{2}}$ .

$\frac{x^{2} - 12 x - 1}{{(x - 6)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{x^{2} - 12 x - 1}{{(x - 6)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{x^{2} - 12 x - 1}{{(x - 6)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$x^{2} - 12 x - 1 = 0$

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 $a = 1$ , $b = - 12$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (1 \cdot - 1)}}{2 \cdot 1}$

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

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

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 $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot - 1}}{2 \cdot 1}$

Step 2.3.3.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{12 \pm \sqrt{144 + 4}}{2 \cdot 1}$

Step 2.3.3.1.3

Add $144$ and $4$ .

$x = \frac{12 \pm \sqrt{148}}{2 \cdot 1}$

Step 2.3.3.1.4

Rewrite $148$ as $2^{2} \cdot 37$ .

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Step 2.3.3.1.4.1

Factor $4$ out of $148$ .

$x = \frac{12 \pm \sqrt{4 (37)}}{2 \cdot 1}$

Step 2.3.3.1.4.2

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

$x = \frac{12 \pm \sqrt{2^{2} \cdot 37}}{2 \cdot 1}$

Step 2.3.3.1.5

Pull terms out from under the radical.

$x = \frac{12 \pm 2 \sqrt{37}}{2 \cdot 1}$

Step 2.3.3.2

Multiply $2$ by $1$ .

$x = \frac{12 \pm 2 \sqrt{37}}{2}$

Step 2.3.3.3

Simplify $\frac{12 \pm 2 \sqrt{37}}{2}$ .

$x = 6 \pm \sqrt{37}$

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

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

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 $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot - 1}}{2 \cdot 1}$

Step 2.3.4.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{12 \pm \sqrt{144 + 4}}{2 \cdot 1}$

Step 2.3.4.1.3

Add $144$ and $4$ .

$x = \frac{12 \pm \sqrt{148}}{2 \cdot 1}$

Step 2.3.4.1.4

Rewrite $148$ as $2^{2} \cdot 37$ .

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Step 2.3.4.1.4.1

Factor $4$ out of $148$ .

$x = \frac{12 \pm \sqrt{4 (37)}}{2 \cdot 1}$

Step 2.3.4.1.4.2

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

$x = \frac{12 \pm \sqrt{2^{2} \cdot 37}}{2 \cdot 1}$

Step 2.3.4.1.5

Pull terms out from under the radical.

$x = \frac{12 \pm 2 \sqrt{37}}{2 \cdot 1}$

Step 2.3.4.2

Multiply $2$ by $1$ .

$x = \frac{12 \pm 2 \sqrt{37}}{2}$

Step 2.3.4.3

Simplify $\frac{12 \pm 2 \sqrt{37}}{2}$ .

$x = 6 \pm \sqrt{37}$

Step 2.3.4.4

Change the $\pm$ to $+$ .

$x = 6 + \sqrt{37}$

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

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

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 $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot - 1}}{2 \cdot 1}$

Step 2.3.5.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{12 \pm \sqrt{144 + 4}}{2 \cdot 1}$

Step 2.3.5.1.3

Add $144$ and $4$ .

$x = \frac{12 \pm \sqrt{148}}{2 \cdot 1}$

Step 2.3.5.1.4

Rewrite $148$ as $2^{2} \cdot 37$ .

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Step 2.3.5.1.4.1

Factor $4$ out of $148$ .

$x = \frac{12 \pm \sqrt{4 (37)}}{2 \cdot 1}$

Step 2.3.5.1.4.2

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

$x = \frac{12 \pm \sqrt{2^{2} \cdot 37}}{2 \cdot 1}$

Step 2.3.5.1.5

Pull terms out from under the radical.

$x = \frac{12 \pm 2 \sqrt{37}}{2 \cdot 1}$

Step 2.3.5.2

Multiply $2$ by $1$ .

$x = \frac{12 \pm 2 \sqrt{37}}{2}$

Step 2.3.5.3

Simplify $\frac{12 \pm 2 \sqrt{37}}{2}$ .

$x = 6 \pm \sqrt{37}$

Step 2.3.5.4

Change the $\pm$ to $-$ .

$x = 6 - \sqrt{37}$

Step 2.3.6

The final answer is the combination of both solutions.

$x = 6 + \sqrt{37}, 6 - \sqrt{37}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{x^{2} - 12 x - 1}{{(x - 6)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x - 6)}^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Set the $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 3.2.2

Add $6$ to both sides of the equation.

$x = 6$

Step 4

Evaluate $\frac{x^{2} - 6 x + 37}{x - 6}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 6 + \sqrt{37}$ .

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Step 4.1.1

Substitute $6 + \sqrt{37}$ for $x$ .

$\frac{{(6 + \sqrt{37})}^{2} - 6 (6 + \sqrt{37}) + 37}{(6 + \sqrt{37}) - 6}$

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

Rewrite ${(6 + \sqrt{37})}^{2}$ as $(6 + \sqrt{37}) (6 + \sqrt{37})$ .

$\frac{(6 + \sqrt{37}) (6 + \sqrt{37}) - 6 (6 + \sqrt{37}) + 37}{6 + \sqrt{37} - 6}$

Step 4.1.2.1.2

Expand $(6 + \sqrt{37}) (6 + \sqrt{37})$ using the FOIL Method.

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Step 4.1.2.1.2.1

Apply the distributive property.

$\frac{6 (6 + \sqrt{37}) + \sqrt{37} (6 + \sqrt{37}) - 6 (6 + \sqrt{37}) + 37}{6 + \sqrt{37} - 6}$

Step 4.1.2.1.2.2

Apply the distributive property.

$\frac{6 \cdot 6 + 6 \sqrt{37} + \sqrt{37} (6 + \sqrt{37}) - 6 (6 + \sqrt{37}) + 37}{6 + \sqrt{37} - 6}$

Step 4.1.2.1.2.3

Apply the distributive property.

$\frac{6 \cdot 6 + 6 \sqrt{37} + \sqrt{37} \cdot 6 + \sqrt{37} \sqrt{37} - 6 (6 + \sqrt{37}) + 37}{6 + \sqrt{37} - 6}$

Step 4.1.2.1.3

Simplify and combine like terms.

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Step 4.1.2.1.3.1

Simplify each term.

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Step 4.1.2.1.3.1.1

Multiply $6$ by $6$ .

$\frac{36 + 6 \sqrt{37} + \sqrt{37} \cdot 6 + \sqrt{37} \sqrt{37} - 6 (6 + \sqrt{37}) + 37}{6 + \sqrt{37} - 6}$

Step 4.1.2.1.3.1.2

Move $6$ to the left of $\sqrt{37}$ .

$\frac{36 + 6 \sqrt{37} + 6 \cdot \sqrt{37} + \sqrt{37} \sqrt{37} - 6 (6 + \sqrt{37}) + 37}{6 + \sqrt{37} - 6}$

Step 4.1.2.1.3.1.3

Combine using the product rule for radicals.

$\frac{36 + 6 \sqrt{37} + 6 \sqrt{37} + \sqrt{37 \cdot 37} - 6 (6 + \sqrt{37}) + 37}{6 + \sqrt{37} - 6}$

Step 4.1.2.1.3.1.4

Multiply $37$ by $37$ .

$\frac{36 + 6 \sqrt{37} + 6 \sqrt{37} + \sqrt{1369} - 6 (6 + \sqrt{37}) + 37}{6 + \sqrt{37} - 6}$

Step 4.1.2.1.3.1.5

Rewrite $1369$ as $37^{2}$ .

$\frac{36 + 6 \sqrt{37} + 6 \sqrt{37} + \sqrt{37^{2}} - 6 (6 + \sqrt{37}) + 37}{6 + \sqrt{37} - 6}$

Step 4.1.2.1.3.1.6

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

$\frac{36 + 6 \sqrt{37} + 6 \sqrt{37} + 37 - 6 (6 + \sqrt{37}) + 37}{6 + \sqrt{37} - 6}$

Step 4.1.2.1.3.2

Add $36$ and $37$ .

$\frac{73 + 6 \sqrt{37} + 6 \sqrt{37} - 6 (6 + \sqrt{37}) + 37}{6 + \sqrt{37} - 6}$

Step 4.1.2.1.3.3

Add $6 \sqrt{37}$ and $6 \sqrt{37}$ .

$\frac{73 + 12 \sqrt{37} - 6 (6 + \sqrt{37}) + 37}{6 + \sqrt{37} - 6}$

Step 4.1.2.1.4

Apply the distributive property.

$\frac{73 + 12 \sqrt{37} - 6 \cdot 6 - 6 \sqrt{37} + 37}{6 + \sqrt{37} - 6}$

Step 4.1.2.1.5

Multiply $- 6$ by $6$ .

$\frac{73 + 12 \sqrt{37} - 36 - 6 \sqrt{37} + 37}{6 + \sqrt{37} - 6}$

Step 4.1.2.1.6

Subtract $36$ from $73$ .

$\frac{37 + 12 \sqrt{37} - 6 \sqrt{37} + 37}{6 + \sqrt{37} - 6}$

Step 4.1.2.1.7

Add $37$ and $37$ .

$\frac{74 + 12 \sqrt{37} - 6 \sqrt{37}}{6 + \sqrt{37} - 6}$

Step 4.1.2.1.8

Subtract $6 \sqrt{37}$ from $12 \sqrt{37}$ .

$\frac{74 + 6 \sqrt{37}}{6 + \sqrt{37} - 6}$

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

Subtract $6$ from $6$ .

$\frac{74 + 6 \sqrt{37}}{0 + \sqrt{37}}$

Step 4.1.2.2.2

Add $0$ and $\sqrt{37}$ .

$\frac{74 + 6 \sqrt{37}}{\sqrt{37}}$

Step 4.1.2.3

Multiply $\frac{74 + 6 \sqrt{37}}{\sqrt{37}}$ by $\frac{\sqrt{37}}{\sqrt{37}}$ .

$\frac{74 + 6 \sqrt{37}}{\sqrt{37}} \cdot \frac{\sqrt{37}}{\sqrt{37}}$

Step 4.1.2.4

Combine and simplify the denominator.

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Step 4.1.2.4.1

Multiply $\frac{74 + 6 \sqrt{37}}{\sqrt{37}}$ by $\frac{\sqrt{37}}{\sqrt{37}}$ .

$\frac{(74 + 6 \sqrt{37}) \sqrt{37}}{\sqrt{37} \sqrt{37}}$

Step 4.1.2.4.2

Raise $\sqrt{37}$ to the power of $1$ .

$\frac{(74 + 6 \sqrt{37}) \sqrt{37}}{{\sqrt{37}}^{1} \sqrt{37}}$

Step 4.1.2.4.3

Raise $\sqrt{37}$ to the power of $1$ .

$\frac{(74 + 6 \sqrt{37}) \sqrt{37}}{{\sqrt{37}}^{1} {\sqrt{37}}^{1}}$

Step 4.1.2.4.4

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

$\frac{(74 + 6 \sqrt{37}) \sqrt{37}}{{\sqrt{37}}^{1 + 1}}$

Step 4.1.2.4.5

Add $1$ and $1$ .

$\frac{(74 + 6 \sqrt{37}) \sqrt{37}}{{\sqrt{37}}^{2}}$

Step 4.1.2.4.6

Rewrite ${\sqrt{37}}^{2}$ as $37$ .

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Step 4.1.2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{37}$ as $37^{\frac{1}{2}}$ .

$\frac{(74 + 6 \sqrt{37}) \sqrt{37}}{{(37^{\frac{1}{2}})}^{2}}$

Step 4.1.2.4.6.2

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

$\frac{(74 + 6 \sqrt{37}) \sqrt{37}}{37^{\frac{1}{2} \cdot 2}}$

Step 4.1.2.4.6.3

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

$\frac{(74 + 6 \sqrt{37}) \sqrt{37}}{37^{\frac{2}{2}}}$

Step 4.1.2.4.6.4

Cancel the common factor of $2$ .

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Step 4.1.2.4.6.4.1

Cancel the common factor.

$\frac{(74 + 6 \sqrt{37}) \sqrt{37}}{37^{\frac{2}{2}}}$

Step 4.1.2.4.6.4.2

Rewrite the expression.

$\frac{(74 + 6 \sqrt{37}) \sqrt{37}}{37^{1}}$

Step 4.1.2.4.6.5

Evaluate the exponent.

$\frac{(74 + 6 \sqrt{37}) \sqrt{37}}{37}$

Step 4.1.2.5

Apply the distributive property.

$\frac{74 \sqrt{37} + 6 \sqrt{37} \sqrt{37}}{37}$

Step 4.1.2.6

Multiply $6 \sqrt{37} \sqrt{37}$ .

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Step 4.1.2.6.1

Raise $\sqrt{37}$ to the power of $1$ .

$\frac{74 \sqrt{37} + 6 ({\sqrt{37}}^{1} \sqrt{37})}{37}$

Step 4.1.2.6.2

Raise $\sqrt{37}$ to the power of $1$ .

$\frac{74 \sqrt{37} + 6 ({\sqrt{37}}^{1} {\sqrt{37}}^{1})}{37}$

Step 4.1.2.6.3

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

$\frac{74 \sqrt{37} + 6 {\sqrt{37}}^{1 + 1}}{37}$

Step 4.1.2.6.4

Add $1$ and $1$ .

$\frac{74 \sqrt{37} + 6 {\sqrt{37}}^{2}}{37}$

Step 4.1.2.7

Simplify each term.

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Step 4.1.2.7.1

Rewrite ${\sqrt{37}}^{2}$ as $37$ .

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Step 4.1.2.7.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{37}$ as $37^{\frac{1}{2}}$ .

$\frac{74 \sqrt{37} + 6 {(37^{\frac{1}{2}})}^{2}}{37}$

Step 4.1.2.7.1.2

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

$\frac{74 \sqrt{37} + 6 \cdot 37^{\frac{1}{2} \cdot 2}}{37}$

Step 4.1.2.7.1.3

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

$\frac{74 \sqrt{37} + 6 \cdot 37^{\frac{2}{2}}}{37}$

Step 4.1.2.7.1.4

Cancel the common factor of $2$ .

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Step 4.1.2.7.1.4.1

Cancel the common factor.

$\frac{74 \sqrt{37} + 6 \cdot 37^{\frac{2}{2}}}{37}$

Step 4.1.2.7.1.4.2

Rewrite the expression.

$\frac{74 \sqrt{37} + 6 \cdot 37^{1}}{37}$

Step 4.1.2.7.1.5

Evaluate the exponent.

$\frac{74 \sqrt{37} + 6 \cdot 37}{37}$

Step 4.1.2.7.2

Multiply $6$ by $37$ .

$\frac{74 \sqrt{37} + 222}{37}$

Step 4.1.2.8

Cancel the common factor of $74 \sqrt{37} + 222$ and $37$ .

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Step 4.1.2.8.1

Factor $37$ out of $74 \sqrt{37}$ .

$\frac{37 (2 \sqrt{37}) + 222}{37}$

Step 4.1.2.8.2

Factor $37$ out of $222$ .

$\frac{37 (2 \sqrt{37}) + 37 \cdot 6}{37}$

Step 4.1.2.8.3

Factor $37$ out of $37 (2 \sqrt{37}) + 37 (6)$ .

$\frac{37 (2 \sqrt{37} + 6)}{37}$

Step 4.1.2.8.4

Cancel the common factors.

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Step 4.1.2.8.4.1

Factor $37$ out of $37$ .

$\frac{37 (2 \sqrt{37} + 6)}{37 (1)}$

Step 4.1.2.8.4.2

Cancel the common factor.

$\frac{37 (2 \sqrt{37} + 6)}{37 \cdot 1}$

Step 4.1.2.8.4.3

Rewrite the expression.

$\frac{2 \sqrt{37} + 6}{1}$

Step 4.1.2.8.4.4

Divide $2 \sqrt{37} + 6$ by $1$ .

$2 \sqrt{37} + 6$

Step 4.2

Evaluate at $x = 6 - \sqrt{37}$ .

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Step 4.2.1

Substitute $6 - \sqrt{37}$ for $x$ .

$\frac{{(6 - \sqrt{37})}^{2} - 6 (6 - \sqrt{37}) + 37}{(6 - \sqrt{37}) - 6}$

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

Rewrite ${(6 - \sqrt{37})}^{2}$ as $(6 - \sqrt{37}) (6 - \sqrt{37})$ .

$\frac{(6 - \sqrt{37}) (6 - \sqrt{37}) - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.2

Expand $(6 - \sqrt{37}) (6 - \sqrt{37})$ using the FOIL Method.

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Step 4.2.2.1.2.1

Apply the distributive property.

$\frac{6 (6 - \sqrt{37}) - \sqrt{37} (6 - \sqrt{37}) - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.2.2

Apply the distributive property.

$\frac{6 \cdot 6 + 6 (- \sqrt{37}) - \sqrt{37} (6 - \sqrt{37}) - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.2.3

Apply the distributive property.

$\frac{6 \cdot 6 + 6 (- \sqrt{37}) - \sqrt{37} \cdot 6 - \sqrt{37} (- \sqrt{37}) - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.3

Simplify and combine like terms.

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Step 4.2.2.1.3.1

Simplify each term.

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Step 4.2.2.1.3.1.1

Multiply $6$ by $6$ .

$\frac{36 + 6 (- \sqrt{37}) - \sqrt{37} \cdot 6 - \sqrt{37} (- \sqrt{37}) - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.3.1.2

Multiply $- 1$ by $6$ .

$\frac{36 - 6 \sqrt{37} - \sqrt{37} \cdot 6 - \sqrt{37} (- \sqrt{37}) - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.3.1.3

Multiply $6$ by $- 1$ .

$\frac{36 - 6 \sqrt{37} - 6 \sqrt{37} - \sqrt{37} (- \sqrt{37}) - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.3.1.4

Multiply $- \sqrt{37} (- \sqrt{37})$ .

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Step 4.2.2.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{36 - 6 \sqrt{37} - 6 \sqrt{37} + 1 \sqrt{37} \sqrt{37} - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.3.1.4.2

Multiply $\sqrt{37}$ by $1$ .

$\frac{36 - 6 \sqrt{37} - 6 \sqrt{37} + \sqrt{37} \sqrt{37} - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.3.1.4.3

Raise $\sqrt{37}$ to the power of $1$ .

$\frac{36 - 6 \sqrt{37} - 6 \sqrt{37} + {\sqrt{37}}^{1} \sqrt{37} - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.3.1.4.4

Raise $\sqrt{37}$ to the power of $1$ .

$\frac{36 - 6 \sqrt{37} - 6 \sqrt{37} + {\sqrt{37}}^{1} {\sqrt{37}}^{1} - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.3.1.4.5

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

$\frac{36 - 6 \sqrt{37} - 6 \sqrt{37} + {\sqrt{37}}^{1 + 1} - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.3.1.4.6

Add $1$ and $1$ .

$\frac{36 - 6 \sqrt{37} - 6 \sqrt{37} + {\sqrt{37}}^{2} - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.3.1.5

Rewrite ${\sqrt{37}}^{2}$ as $37$ .

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Step 4.2.2.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{37}$ as $37^{\frac{1}{2}}$ .

$\frac{36 - 6 \sqrt{37} - 6 \sqrt{37} + {(37^{\frac{1}{2}})}^{2} - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.3.1.5.2

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

$\frac{36 - 6 \sqrt{37} - 6 \sqrt{37} + 37^{\frac{1}{2} \cdot 2} - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.3.1.5.3

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

$\frac{36 - 6 \sqrt{37} - 6 \sqrt{37} + 37^{\frac{2}{2}} - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.2.1.3.1.5.4.1

Cancel the common factor.

$\frac{36 - 6 \sqrt{37} - 6 \sqrt{37} + 37^{\frac{2}{2}} - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.3.1.5.4.2

Rewrite the expression.

$\frac{36 - 6 \sqrt{37} - 6 \sqrt{37} + 37^{1} - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.3.1.5.5

Evaluate the exponent.

$\frac{36 - 6 \sqrt{37} - 6 \sqrt{37} + 37 - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.3.2

Add $36$ and $37$ .

$\frac{73 - 6 \sqrt{37} - 6 \sqrt{37} - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.3.3

Subtract $6 \sqrt{37}$ from $- 6 \sqrt{37}$ .

$\frac{73 - 12 \sqrt{37} - 6 (6 - \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.4

Apply the distributive property.

$\frac{73 - 12 \sqrt{37} - 6 \cdot 6 - 6 (- \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.5

Multiply $- 6$ by $6$ .

$\frac{73 - 12 \sqrt{37} - 36 - 6 (- \sqrt{37}) + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.6

Multiply $- 1$ by $- 6$ .

$\frac{73 - 12 \sqrt{37} - 36 + 6 \sqrt{37} + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.7

Subtract $36$ from $73$ .

$\frac{37 - 12 \sqrt{37} + 6 \sqrt{37} + 37}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.8

Add $37$ and $37$ .

$\frac{74 - 12 \sqrt{37} + 6 \sqrt{37}}{6 - \sqrt{37} - 6}$

Step 4.2.2.1.9

Add $- 12 \sqrt{37}$ and $6 \sqrt{37}$ .

$\frac{74 - 6 \sqrt{37}}{6 - \sqrt{37} - 6}$

Step 4.2.2.2

Simplify the denominator.

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Step 4.2.2.2.1

Subtract $6$ from $6$ .

$\frac{74 - 6 \sqrt{37}}{0 - \sqrt{37}}$

Step 4.2.2.2.2

Subtract $\sqrt{37}$ from $0$ .

$\frac{74 - 6 \sqrt{37}}{- \sqrt{37}}$

Step 4.2.2.3

Move the negative in front of the fraction.

$- \frac{74 - 6 \sqrt{37}}{\sqrt{37}}$

Step 4.2.2.4

Multiply $\frac{74 - 6 \sqrt{37}}{\sqrt{37}}$ by $\frac{\sqrt{37}}{\sqrt{37}}$ .

$- (\frac{74 - 6 \sqrt{37}}{\sqrt{37}} \cdot \frac{\sqrt{37}}{\sqrt{37}})$

Step 4.2.2.5

Combine and simplify the denominator.

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Step 4.2.2.5.1

Multiply $\frac{74 - 6 \sqrt{37}}{\sqrt{37}}$ by $\frac{\sqrt{37}}{\sqrt{37}}$ .

$- \frac{(74 - 6 \sqrt{37}) \sqrt{37}}{\sqrt{37} \sqrt{37}}$

Step 4.2.2.5.2

Raise $\sqrt{37}$ to the power of $1$ .

$- \frac{(74 - 6 \sqrt{37}) \sqrt{37}}{{\sqrt{37}}^{1} \sqrt{37}}$

Step 4.2.2.5.3

Raise $\sqrt{37}$ to the power of $1$ .

$- \frac{(74 - 6 \sqrt{37}) \sqrt{37}}{{\sqrt{37}}^{1} {\sqrt{37}}^{1}}$

Step 4.2.2.5.4

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

$- \frac{(74 - 6 \sqrt{37}) \sqrt{37}}{{\sqrt{37}}^{1 + 1}}$

Step 4.2.2.5.5

Add $1$ and $1$ .

$- \frac{(74 - 6 \sqrt{37}) \sqrt{37}}{{\sqrt{37}}^{2}}$

Step 4.2.2.5.6

Rewrite ${\sqrt{37}}^{2}$ as $37$ .

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Step 4.2.2.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{37}$ as $37^{\frac{1}{2}}$ .

$- \frac{(74 - 6 \sqrt{37}) \sqrt{37}}{{(37^{\frac{1}{2}})}^{2}}$

Step 4.2.2.5.6.2

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

$- \frac{(74 - 6 \sqrt{37}) \sqrt{37}}{37^{\frac{1}{2} \cdot 2}}$

Step 4.2.2.5.6.3

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

$- \frac{(74 - 6 \sqrt{37}) \sqrt{37}}{37^{\frac{2}{2}}}$

Step 4.2.2.5.6.4

Cancel the common factor of $2$ .

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Step 4.2.2.5.6.4.1

Cancel the common factor.

$- \frac{(74 - 6 \sqrt{37}) \sqrt{37}}{37^{\frac{2}{2}}}$

Step 4.2.2.5.6.4.2

Rewrite the expression.

$- \frac{(74 - 6 \sqrt{37}) \sqrt{37}}{37^{1}}$

Step 4.2.2.5.6.5

Evaluate the exponent.

$- \frac{(74 - 6 \sqrt{37}) \sqrt{37}}{37}$

Step 4.2.2.6

Apply the distributive property.

$- \frac{74 \sqrt{37} - 6 \sqrt{37} \sqrt{37}}{37}$

Step 4.2.2.7

Multiply $- 6 \sqrt{37} \sqrt{37}$ .

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Step 4.2.2.7.1

Raise $\sqrt{37}$ to the power of $1$ .

$- \frac{74 \sqrt{37} - 6 ({\sqrt{37}}^{1} \sqrt{37})}{37}$

Step 4.2.2.7.2

Raise $\sqrt{37}$ to the power of $1$ .

$- \frac{74 \sqrt{37} - 6 ({\sqrt{37}}^{1} {\sqrt{37}}^{1})}{37}$

Step 4.2.2.7.3

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

$- \frac{74 \sqrt{37} - 6 {\sqrt{37}}^{1 + 1}}{37}$

Step 4.2.2.7.4

Add $1$ and $1$ .

$- \frac{74 \sqrt{37} - 6 {\sqrt{37}}^{2}}{37}$

Step 4.2.2.8

Simplify each term.

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Step 4.2.2.8.1

Rewrite ${\sqrt{37}}^{2}$ as $37$ .

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Step 4.2.2.8.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{37}$ as $37^{\frac{1}{2}}$ .

$- \frac{74 \sqrt{37} - 6 {(37^{\frac{1}{2}})}^{2}}{37}$

Step 4.2.2.8.1.2

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

$- \frac{74 \sqrt{37} - 6 \cdot 37^{\frac{1}{2} \cdot 2}}{37}$

Step 4.2.2.8.1.3

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

$- \frac{74 \sqrt{37} - 6 \cdot 37^{\frac{2}{2}}}{37}$

Step 4.2.2.8.1.4

Cancel the common factor of $2$ .

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Step 4.2.2.8.1.4.1

Cancel the common factor.

$- \frac{74 \sqrt{37} - 6 \cdot 37^{\frac{2}{2}}}{37}$

Step 4.2.2.8.1.4.2

Rewrite the expression.

$- \frac{74 \sqrt{37} - 6 \cdot 37^{1}}{37}$

Step 4.2.2.8.1.5

Evaluate the exponent.

$- \frac{74 \sqrt{37} - 6 \cdot 37}{37}$

Step 4.2.2.8.2

Multiply $- 6$ by $37$ .

$- \frac{74 \sqrt{37} - 222}{37}$

Step 4.2.2.9

Simplify terms.

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Step 4.2.2.9.1

Cancel the common factor of $74 \sqrt{37} - 222$ and $37$ .

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Step 4.2.2.9.1.1

Factor $37$ out of $74 \sqrt{37}$ .

$- \frac{37 (2 \sqrt{37}) - 222}{37}$

Step 4.2.2.9.1.2

Factor $37$ out of $- 222$ .

$- \frac{37 (2 \sqrt{37}) + 37 \cdot - 6}{37}$

Step 4.2.2.9.1.3

Factor $37$ out of $37 (2 \sqrt{37}) + 37 (- 6)$ .

$- \frac{37 (2 \sqrt{37} - 6)}{37}$

Step 4.2.2.9.1.4

Cancel the common factors.

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Step 4.2.2.9.1.4.1

Factor $37$ out of $37$ .

$- \frac{37 (2 \sqrt{37} - 6)}{37 (1)}$

Step 4.2.2.9.1.4.2

Cancel the common factor.

$- \frac{37 (2 \sqrt{37} - 6)}{37 \cdot 1}$

Step 4.2.2.9.1.4.3

Rewrite the expression.

$- \frac{2 \sqrt{37} - 6}{1}$

Step 4.2.2.9.1.4.4

Divide $2 \sqrt{37} - 6$ by $1$ .

$- (2 \sqrt{37} - 6)$

Step 4.2.2.9.2

Apply the distributive property.

$- (2 \sqrt{37}) - - 6$

Step 4.2.2.9.3

Multiply.

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Step 4.2.2.9.3.1

Multiply $2$ by $- 1$ .

$- 2 \sqrt{37} - - 6$

Step 4.2.2.9.3.2

Multiply $- 1$ by $- 6$ .

$- 2 \sqrt{37} + 6$

Step 4.3

Evaluate at $x = 6$ .

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Step 4.3.1

Substitute $6$ for $x$ .

$\frac{{(6)}^{2} - 6 \cdot 6 + 37}{(6) - 6}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Subtract $6$ from $6$ .

$\frac{{(6)}^{2} - 6 \cdot 6 + 37}{0}$

Step 4.3.2.2

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

Undefined

Step 4.4

List all of the points.

$(6 + \sqrt{37}, 2 \sqrt{37} + 6), (6 - \sqrt{37}, - 2 \sqrt{37} + 6)$

Step 5