Precalculus Examples

$\frac{x + 2}{x^{2} + 5 x + 6}$

Step 1

Write $\frac{x + 2}{x^{2} + 5 x + 6}$ as an equation.

$y = \frac{x + 2}{x^{2} + 5 x + 6}$

Step 2

Find the x-intercepts.

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Step 2.1

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

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

Step 2.2

Solve the equation.

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Step 2.2.1

Set the numerator equal to zero.

$x + 2 = 0$

Step 2.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.2.3

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

$No solution$

Step 2.3

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

x-intercept(s): $None$

Step 3

Find the y-intercepts.

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Step 3.1

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

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

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

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

Step 3.2.2

Remove parentheses.

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

Step 3.2.3

Remove parentheses.

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

Step 3.2.4

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

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Step 3.2.4.1

Add $0$ and $2$ .

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

Step 3.2.4.2

Simplify the denominator.

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Step 3.2.4.2.1

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

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

Step 3.2.4.2.2

Multiply $5$ by $0$ .

$y = \frac{2}{0 + 0 + 6}$

Step 3.2.4.2.3

Add $0$ and $0$ .

$y = \frac{2}{0 + 6}$

Step 3.2.4.2.4

Add $0$ and $6$ .

$y = \frac{2}{6}$

Step 3.2.4.3

Cancel the common factor of $2$ and $6$ .

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Step 3.2.4.3.1

Factor $2$ out of $2$ .

$y = \frac{2 (1)}{6}$

Step 3.2.4.3.2

Cancel the common factors.

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Step 3.2.4.3.2.1

Factor $2$ out of $6$ .

$y = \frac{2 \cdot 1}{2 \cdot 3}$

Step 3.2.4.3.2.2

Cancel the common factor.

$y = \frac{2 \cdot 1}{2 \cdot 3}$

Step 3.2.4.3.2.3

Rewrite the expression.

$y = \frac{1}{3}$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{1}{3})$

Step 4

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, \frac{1}{3})$

Step 5