Precalculus Examples

$f (x) = \frac{(x + 2) (x - 2)}{3 x^{2} + 8 x + 4}$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{(x + 2) (x - 2)}{3 x^{2} + 8 x + 4}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$(x + 2) (x - 2) = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

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

$x + 2 = 0$

$x - 2 = 0$

Step 1.2.2.2

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 1.2.2.2.1

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

$x + 2 = 0$

Step 1.2.2.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.2.3

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 1.2.2.3.1

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

$x - 2 = 0$

Step 1.2.2.3.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.2.4

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

$x = - 2, 2$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{(x + 2) (x - 2)}{3 x^{2} + 8 x + 4}$ true.

$x = 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(2, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{((0) + 2) ((0) - 2)}{3 {(0)}^{2} + 8 (0) + 4}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{(0 + 2) ((0) - 2)}{3 {(0)}^{2} + 8 (0) + 4}$

Step 2.2.2

Remove parentheses.

$y = \frac{(0 + 2) (0 - 2)}{3 {(0)}^{2} + 8 (0) + 4}$

Step 2.2.3

Remove parentheses.

$y = \frac{(0 + 2) (0 - 2)}{3 \cdot 0^{2} + 8 (0) + 4}$

Step 2.2.4

Remove parentheses.

$y = \frac{((0) + 2) ((0) - 2)}{3 {(0)}^{2} + 8 (0) + 4}$

Step 2.2.5

Simplify $\frac{((0) + 2) ((0) - 2)}{3 {(0)}^{2} + 8 (0) + 4}$ .

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Step 2.2.5.1

Simplify the numerator.

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Step 2.2.5.1.1

Rewrite $0$ as $- 1 (0)$ .

$y = \frac{(- 1 (0) + 2) (0 - 2)}{3 \cdot 0^{2} + 8 (0) + 4}$

Step 2.2.5.1.2

Rewrite $2$ as $- 1 (- 2)$ .

$y = \frac{(- 1 \cdot 0 - 1 \cdot - 2) (0 - 2)}{3 \cdot 0^{2} + 8 (0) + 4}$

Step 2.2.5.1.3

Factor $- 1$ out of $- 1 \cdot 0 - 1 \cdot - 2$ .

$y = \frac{- 1 \cdot (0 - 2) (0 - 2)}{3 \cdot 0^{2} + 8 (0) + 4}$

Step 2.2.5.1.4

Raise $0 - 2$ to the power of $1$ .

$y = \frac{- 1 \cdot ({(0 - 2)}^{1} (0 - 2))}{3 \cdot 0^{2} + 8 (0) + 4}$

Step 2.2.5.1.5

Raise $0 - 2$ to the power of $1$ .

$y = \frac{- 1 \cdot ({(0 - 2)}^{1} {(0 - 2)}^{1})}{3 \cdot 0^{2} + 8 (0) + 4}$

Step 2.2.5.1.6

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

$y = \frac{- 1 \cdot {(0 - 2)}^{1 + 1}}{3 \cdot 0^{2} + 8 (0) + 4}$

Step 2.2.5.1.7

Add $1$ and $1$ .

$y = \frac{- 1 \cdot {(0 - 2)}^{2}}{3 \cdot 0^{2} + 8 (0) + 4}$

Step 2.2.5.2

Simplify the denominator.

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Step 2.2.5.2.1

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

$y = \frac{- 1 \cdot {(0 - 2)}^{2}}{3 \cdot 0 + 8 (0) + 4}$

Step 2.2.5.2.2

Multiply $3$ by $0$ .

$y = \frac{- 1 \cdot {(0 - 2)}^{2}}{0 + 8 (0) + 4}$

Step 2.2.5.2.3

Multiply $8$ by $0$ .

$y = \frac{- 1 \cdot {(0 - 2)}^{2}}{0 + 0 + 4}$

Step 2.2.5.2.4

Add $0$ and $0$ .

$y = \frac{- 1 \cdot {(0 - 2)}^{2}}{0 + 4}$

Step 2.2.5.2.5

Add $0$ and $4$ .

$y = \frac{- 1 \cdot {(0 - 2)}^{2}}{4}$

Step 2.2.5.3

Simplify the numerator.

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Step 2.2.5.3.1

Subtract $2$ from $0$ .

$y = \frac{- 1 {(- 2)}^{2}}{4}$

Step 2.2.5.3.2

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

$y = \frac{- 1 \cdot 4}{4}$

Step 2.2.5.4

Simplify the expression.

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Step 2.2.5.4.1

Multiply $- 1$ by $4$ .

$y = \frac{- 4}{4}$

Step 2.2.5.4.2

Divide $- 4$ by $4$ .

$y = - 1$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 1)$

Step 3

List the intersections.

x-intercept(s): $(2, 0)$

y-intercept(s): $(0, - 1)$

Step 4