Calculus Examples

$f (x) = \frac{3 x^{2} + 5 x - 12}{x^{2} - 4}$

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 $f (x) = 3 x^{2} + 5 x - 12$ and $g (x) = x^{2} - 4$ .

$\frac{(x^{2} - 4) \frac{d}{d x} [3 x^{2} + 5 x - 12] - (3 x^{2} + 5 x - 12) \frac{d}{d x} [x^{2} - 4]}{{(x^{2} - 4)}^{2}}$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $3 x^{2} + 5 x - 12$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 12]$ .

$\frac{(x^{2} - 4) (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 12]) - (3 x^{2} + 5 x - 12) \frac{d}{d x} [x^{2} - 4]}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$\frac{(x^{2} - 4) (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 12]) - (3 x^{2} + 5 x - 12) \frac{d}{d x} [x^{2} - 4]}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.3

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

$\frac{(x^{2} - 4) (3 (2 x) + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 12]) - (3 x^{2} + 5 x - 12) \frac{d}{d x} [x^{2} - 4]}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.4

Multiply $2$ by $3$ .

$\frac{(x^{2} - 4) (6 x + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 12]) - (3 x^{2} + 5 x - 12) \frac{d}{d x} [x^{2} - 4]}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.5

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

$\frac{(x^{2} - 4) (6 x + 5 \frac{d}{d x} [x] + \frac{d}{d x} [- 12]) - (3 x^{2} + 5 x - 12) \frac{d}{d x} [x^{2} - 4]}{{(x^{2} - 4)}^{2}}$

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 $n = 1$ .

$\frac{(x^{2} - 4) (6 x + 5 \cdot 1 + \frac{d}{d x} [- 12]) - (3 x^{2} + 5 x - 12) \frac{d}{d x} [x^{2} - 4]}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.7

Multiply $5$ by $1$ .

$\frac{(x^{2} - 4) (6 x + 5 + \frac{d}{d x} [- 12]) - (3 x^{2} + 5 x - 12) \frac{d}{d x} [x^{2} - 4]}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.8

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12$ with respect to $x$ is $0$ .

$\frac{(x^{2} - 4) (6 x + 5 + 0) - (3 x^{2} + 5 x - 12) \frac{d}{d x} [x^{2} - 4]}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.9

Add $6 x + 5$ and $0$ .

$\frac{(x^{2} - 4) (6 x + 5) - (3 x^{2} + 5 x - 12) \frac{d}{d x} [x^{2} - 4]}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.10

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

$\frac{(x^{2} - 4) (6 x + 5) - (3 x^{2} + 5 x - 12) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4])}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.11

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

$\frac{(x^{2} - 4) (6 x + 5) - (3 x^{2} + 5 x - 12) (2 x + \frac{d}{d x} [- 4])}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.12

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$\frac{(x^{2} - 4) (6 x + 5) - (3 x^{2} + 5 x - 12) (2 x + 0)}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.13

Simplify the expression.

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Step 1.1.2.13.1

Add $2 x$ and $0$ .

$\frac{(x^{2} - 4) (6 x + 5) - (3 x^{2} + 5 x - 12) (2 x)}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.13.2

Multiply $2$ by $- 1$ .

$\frac{(x^{2} - 4) (6 x + 5) - 2 (3 x^{2} + 5 x - 12) x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$\frac{(x^{2} - 4) (6 x + 5) + (- 2 (3 x^{2}) - 2 (5 x) - 2 \cdot - 12) x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.2

Apply the distributive property.

$\frac{(x^{2} - 4) (6 x + 5) - 2 (3 x^{2}) x - 2 (5 x) x - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3

Simplify the numerator.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Expand $(x^{2} - 4) (6 x + 5)$ using the FOIL Method.

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Step 1.1.3.3.1.1.1

Apply the distributive property.

$\frac{x^{2} (6 x + 5) - 4 (6 x + 5) - 2 (3 x^{2}) x - 2 (5 x) x - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.1.2

Apply the distributive property.

$\frac{x^{2} (6 x) + x^{2} \cdot 5 - 4 (6 x + 5) - 2 (3 x^{2}) x - 2 (5 x) x - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.1.3

Apply the distributive property.

$\frac{x^{2} (6 x) + x^{2} \cdot 5 - 4 (6 x) - 4 \cdot 5 - 2 (3 x^{2}) x - 2 (5 x) x - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.2

Simplify each term.

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Step 1.1.3.3.1.2.1

Rewrite using the commutative property of multiplication.

$\frac{6 x^{2} x + x^{2} \cdot 5 - 4 (6 x) - 4 \cdot 5 - 2 (3 x^{2}) x - 2 (5 x) x - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.2.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.1.3.3.1.2.2.1

Move $x$ .

$\frac{6 (x \cdot x^{2}) + x^{2} \cdot 5 - 4 (6 x) - 4 \cdot 5 - 2 (3 x^{2}) x - 2 (5 x) x - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.2.2.2

Multiply $x$ by $x^{2}$ .

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Step 1.1.3.3.1.2.2.2.1

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

$\frac{6 (x^{1} x^{2}) + x^{2} \cdot 5 - 4 (6 x) - 4 \cdot 5 - 2 (3 x^{2}) x - 2 (5 x) x - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.2.2.2.2

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

$\frac{6 x^{1 + 2} + x^{2} \cdot 5 - 4 (6 x) - 4 \cdot 5 - 2 (3 x^{2}) x - 2 (5 x) x - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.2.2.3

Add $1$ and $2$ .

$\frac{6 x^{3} + x^{2} \cdot 5 - 4 (6 x) - 4 \cdot 5 - 2 (3 x^{2}) x - 2 (5 x) x - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.2.3

Move $5$ to the left of $x^{2}$ .

$\frac{6 x^{3} + 5 \cdot x^{2} - 4 (6 x) - 4 \cdot 5 - 2 (3 x^{2}) x - 2 (5 x) x - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.2.4

Multiply $6$ by $- 4$ .

$\frac{6 x^{3} + 5 x^{2} - 24 x - 4 \cdot 5 - 2 (3 x^{2}) x - 2 (5 x) x - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.2.5

Multiply $- 4$ by $5$ .

$\frac{6 x^{3} + 5 x^{2} - 24 x - 20 - 2 (3 x^{2}) x - 2 (5 x) x - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.1.3.3.1.3.1

Move $x$ .

$\frac{6 x^{3} + 5 x^{2} - 24 x - 20 - 2 (3 (x \cdot x^{2})) - 2 (5 x) x - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.3.2

Multiply $x$ by $x^{2}$ .

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Step 1.1.3.3.1.3.2.1

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

$\frac{6 x^{3} + 5 x^{2} - 24 x - 20 - 2 (3 (x^{1} x^{2})) - 2 (5 x) x - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.3.2.2

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

$\frac{6 x^{3} + 5 x^{2} - 24 x - 20 - 2 (3 x^{1 + 2}) - 2 (5 x) x - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.3.3

Add $1$ and $2$ .

$\frac{6 x^{3} + 5 x^{2} - 24 x - 20 - 2 (3 x^{3}) - 2 (5 x) x - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.4

Multiply $3$ by $- 2$ .

$\frac{6 x^{3} + 5 x^{2} - 24 x - 20 - 6 x^{3} - 2 (5 x) x - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.5

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

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Step 1.1.3.3.1.5.1

Move $x$ .

$\frac{6 x^{3} + 5 x^{2} - 24 x - 20 - 6 x^{3} - 2 (5 (x \cdot x)) - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.5.2

Multiply $x$ by $x$ .

$\frac{6 x^{3} + 5 x^{2} - 24 x - 20 - 6 x^{3} - 2 (5 x^{2}) - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.6

Multiply $5$ by $- 2$ .

$\frac{6 x^{3} + 5 x^{2} - 24 x - 20 - 6 x^{3} - 10 x^{2} - 2 \cdot - 12 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.1.7

Multiply $- 2$ by $- 12$ .

$\frac{6 x^{3} + 5 x^{2} - 24 x - 20 - 6 x^{3} - 10 x^{2} + 24 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.2

Combine the opposite terms in $6 x^{3} + 5 x^{2} - 24 x - 20 - 6 x^{3} - 10 x^{2} + 24 x$ .

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Step 1.1.3.3.2.1

Subtract $6 x^{3}$ from $6 x^{3}$ .

$\frac{5 x^{2} - 24 x - 20 + 0 - 10 x^{2} + 24 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.2.2

Add $5 x^{2} - 24 x - 20$ and $0$ .

$\frac{5 x^{2} - 24 x - 20 - 10 x^{2} + 24 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.2.3

Add $- 24 x$ and $24 x$ .

$\frac{5 x^{2} - 20 - 10 x^{2} + 0}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.2.4

Add $5 x^{2} - 20 - 10 x^{2}$ and $0$ .

$\frac{5 x^{2} - 20 - 10 x^{2}}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.3.3

Subtract $10 x^{2}$ from $5 x^{2}$ .

$\frac{- 5 x^{2} - 20}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.4

Factor $5$ out of $- 5 x^{2} - 20$ .

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Step 1.1.3.4.1

Factor $5$ out of $- 5 x^{2}$ .

$\frac{5 (- x^{2}) - 20}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.4.2

Factor $5$ out of $- 20$ .

$\frac{5 (- x^{2}) + 5 (- 4)}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.4.3

Factor $5$ out of $5 (- x^{2}) + 5 (- 4)$ .

$\frac{5 (- x^{2} - 4)}{{(x^{2} - 4)}^{2}}$

Step 1.1.3.5

Simplify the denominator.

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Step 1.1.3.5.1

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

$\frac{5 (- x^{2} - 4)}{{(x^{2} - 2^{2})}^{2}}$

Step 1.1.3.5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{5 (- x^{2} - 4)}{{((x + 2) (x - 2))}^{2}}$

Step 1.1.3.5.3

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

$\frac{5 (- x^{2} - 4)}{{(x + 2)}^{2} {(x - 2)}^{2}}$

Step 1.1.3.6

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

$\frac{5 (- (x^{2}) - 4)}{{(x + 2)}^{2} {(x - 2)}^{2}}$

Step 1.1.3.7

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{5 (- (x^{2}) - 1 (4))}{{(x + 2)}^{2} {(x - 2)}^{2}}$

Step 1.1.3.8

Factor $- 1$ out of $- (x^{2}) - 1 (4)$ .

$\frac{5 (- (x^{2} + 4))}{{(x + 2)}^{2} {(x - 2)}^{2}}$

Step 1.1.3.9

Rewrite $- (x^{2} + 4)$ as $- 1 (x^{2} + 4)$ .

$\frac{5 (- 1 (x^{2} + 4))}{{(x + 2)}^{2} {(x - 2)}^{2}}$

Step 1.1.3.10

Move the negative in front of the fraction.

$f' (x) = - \frac{5 (x^{2} + 4)}{{(x + 2)}^{2} {(x - 2)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{5 (x^{2} + 4)}{{(x + 2)}^{2} {(x - 2)}^{2}}$ .

$- \frac{5 (x^{2} + 4)}{{(x + 2)}^{2} {(x - 2)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{5 (x^{2} + 4)}{{(x + 2)}^{2} {(x - 2)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{5 (x^{2} + 4)}{{(x + 2)}^{2} {(x - 2)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$5 (x^{2} + 4) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Divide each term in $5 (x^{2} + 4) = 0$ by $5$ and simplify.

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Step 2.3.1.1

Divide each term in $5 (x^{2} + 4) = 0$ by $5$ .

$\frac{5 (x^{2} + 4)}{5} = \frac{0}{5}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $5$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{5 (x^{2} + 4)}{5} = \frac{0}{5}$

Step 2.3.1.2.1.2

Divide $x^{2} + 4$ by $1$ .

$x^{2} + 4 = \frac{0}{5}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $5$ .

$x^{2} + 4 = 0$

Step 2.3.2

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

Step 2.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

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

Step 2.3.4

Simplify $\pm \sqrt{- 4}$ .

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Step 2.3.4.1

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Step 2.3.4.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 2.3.4.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{4}$

Step 2.3.4.4

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

$x = \pm i \cdot \sqrt{2^{2}}$

Step 2.3.4.5

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

$x = \pm i \cdot 2$

Step 2.3.4.6

Move $2$ to the left of $i$ .

$x = \pm 2 i$

Step 2.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2 i$

Step 2.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2 i$

Step 2.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2 i, - 2 i$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{5 (x^{2} + 4)}{{(x + 2)}^{2} {(x - 2)}^{2}}$ equal to $0$ to find where the expression is undefined.

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

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 $0$ .

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

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

Step 3.2.2

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

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Step 3.2.2.1

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

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

Step 3.2.2.2

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

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Step 3.2.2.2.1

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

$x + 2 = 0$

Step 3.2.2.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

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

$x = - 2, 2$

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 $0$ .

$x = - 2, x = 2$

Step 4

Evaluate $\frac{3 x^{2} + 5 x - 12}{x^{2} - 4}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - 2$ .

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Step 4.1.1

Substitute $- 2$ for $x$ .

$\frac{3 {(- 2)}^{2} + 5 (- 2) - 12}{{(- 2)}^{2} - 4}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

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

$\frac{3 {(- 2)}^{2} + 5 (- 2) - 12}{4 - 4}$

Step 4.1.2.2

Subtract $4$ from $4$ .

$\frac{3 {(- 2)}^{2} + 5 (- 2) - 12}{0}$

Step 4.1.2.3

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

Undefined

Step 4.2

Evaluate at $x = 2$ .

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Step 4.2.1

Substitute $2$ for $x$ .

$\frac{3 {(2)}^{2} + 5 (2) - 12}{{(2)}^{2} - 4}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

$\frac{3 {(2)}^{2} + 5 (2) - 12}{4 - 4}$

Step 4.2.2.2

Subtract $4$ from $4$ .

$\frac{3 {(2)}^{2} + 5 (2) - 12}{0}$

Step 4.2.2.3

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

Undefined

Step 5

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found