Precalculus Examples

$f (x) = \frac{x^{2} + 5 x + 6}{x^{2} - 9}$

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 $x$ .

$0 = \frac{x^{2} + 5 x + 6}{x^{2} - 9}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$x^{2} + 5 x + 6 = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

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

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Step 1.2.2.1.1

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 $5$ .

$2, 3$

Step 1.2.2.1.2

Write the factored form using these integers.

$(x + 2) (x + 3) = 0$

Step 1.2.2.2

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

$x + 2 = 0$

$x + 3 = 0$

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

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.2.4

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 1.2.2.4.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 1.2.2.4.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 1.2.2.5

The final solution is all the values that make $(x + 2) (x + 3) = 0$ true.

$x = - 2, - 3$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{x^{2} + 5 x + 6}{x^{2} - 9}$ 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 $y$ .

$y = \frac{{(0)}^{2} + 5 (0) + 6}{{(0)}^{2} - 9}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{0^{2} + 5 (0) + 6}{{(0)}^{2} - 9}$

Step 2.2.2

Remove parentheses.

$y = \frac{0^{2} + 5 (0) + 6}{0^{2} - 9}$

Step 2.2.3

Remove parentheses.

$y = \frac{{(0)}^{2} + 5 (0) + 6}{{(0)}^{2} - 9}$

Step 2.2.4

Simplify $\frac{{(0)}^{2} + 5 (0) + 6}{{(0)}^{2} - 9}$ .

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Step 2.2.4.1

Simplify the numerator.

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Step 2.2.4.1.1

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

$y = \frac{0 + 5 (0) + 6}{0^{2} - 9}$

Step 2.2.4.1.2

Multiply $5$ by $0$ .

$y = \frac{0 + 0 + 6}{0^{2} - 9}$

Step 2.2.4.1.3

Add $0$ and $0$ .

$y = \frac{0 + 6}{0^{2} - 9}$

Step 2.2.4.1.4

Add $0$ and $6$ .

$y = \frac{6}{0^{2} - 9}$

Step 2.2.4.2

Simplify the denominator.

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Step 2.2.4.2.1

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

$y = \frac{6}{0 - 9}$

Step 2.2.4.2.2

Subtract $9$ from $0$ .

$y = \frac{6}{- 9}$

Step 2.2.4.3

Reduce the expression by cancelling the common factors.

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Step 2.2.4.3.1

Cancel the common factor of $6$ and $- 9$ .

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Step 2.2.4.3.1.1

Factor $3$ out of $6$ .

$y = \frac{3 (2)}{- 9}$

Step 2.2.4.3.1.2

Cancel the common factors.

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Step 2.2.4.3.1.2.1

Factor $3$ out of $- 9$ .

$y = \frac{3 \cdot 2}{3 \cdot - 3}$

Step 2.2.4.3.1.2.2

Cancel the common factor.

$y = \frac{3 \cdot 2}{3 \cdot - 3}$

Step 2.2.4.3.1.2.3

Rewrite the expression.

$y = \frac{2}{- 3}$

Step 2.2.4.3.2

Move the negative in front of the fraction.

$y = - \frac{2}{3}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{2}{3})$

Step 3

List the intersections.

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

y-intercept(s): $(0, - \frac{2}{3})$

Step 4