Precalculus Examples

$\frac{5 x + 20}{x^{2} + x - 12}$

Step 1

Write $\frac{5 x + 20}{x^{2} + x - 12}$ as an equation.

$y = \frac{5 x + 20}{x^{2} + x - 12}$

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 $x$ .

$0 = \frac{5 x + 20}{x^{2} + x - 12}$

Step 2.2

Solve the equation.

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Step 2.2.1

Set the numerator equal to zero.

$5 x + 20 = 0$

Step 2.2.2

Solve the equation for $x$ .

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Step 2.2.2.1

Subtract $20$ from both sides of the equation.

$5 x = - 20$

Step 2.2.2.2

Divide each term in $5 x = - 20$ by $5$ and simplify.

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Step 2.2.2.2.1

Divide each term in $5 x = - 20$ by $5$ .

$\frac{5 x}{5} = \frac{- 20}{5}$

Step 2.2.2.2.2

Simplify the left side.

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Step 2.2.2.2.2.1

Cancel the common factor of $5$ .

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Step 2.2.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 20}{5}$

Step 2.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 20}{5}$

Step 2.2.2.2.3

Simplify the right side.

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Step 2.2.2.2.3.1

Divide $- 20$ by $5$ .

$x = - 4$

Step 2.2.3

Exclude the solutions that do not make $0 = \frac{5 x + 20}{x^{2} + x - 12}$ 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 $y$ .

$y = \frac{5 (0) + 20}{{(0)}^{2} + 0 - 12}$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = \frac{5 (0) + 20}{{(0)}^{2} + 0 - 12}$

Step 3.2.2

Remove parentheses.

$y = \frac{5 (0) + 20}{0^{2} + 0 - 12}$

Step 3.2.3

Remove parentheses.

$y = \frac{5 (0) + 20}{{(0)}^{2} + 0 - 12}$

Step 3.2.4

Simplify $\frac{5 (0) + 20}{{(0)}^{2} + 0 - 12}$ .

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Step 3.2.4.1

Simplify the numerator.

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Step 3.2.4.1.1

Multiply $5$ by $0$ .

$y = \frac{0 + 20}{0^{2} + 0 - 12}$

Step 3.2.4.1.2

Add $0$ and $20$ .

$y = \frac{20}{0^{2} + 0 - 12}$

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 $0$ .

$y = \frac{20}{0 + 0 - 12}$

Step 3.2.4.2.2

Add $0$ and $0$ .

$y = \frac{20}{0 - 12}$

Step 3.2.4.2.3

Subtract $12$ from $0$ .

$y = \frac{20}{- 12}$

Step 3.2.4.3

Reduce the expression by cancelling the common factors.

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Step 3.2.4.3.1

Cancel the common factor of $20$ and $- 12$ .

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Step 3.2.4.3.1.1

Factor $4$ out of $20$ .

$y = \frac{4 (5)}{- 12}$

Step 3.2.4.3.1.2

Cancel the common factors.

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Step 3.2.4.3.1.2.1

Factor $4$ out of $- 12$ .

$y = \frac{4 \cdot 5}{4 \cdot - 3}$

Step 3.2.4.3.1.2.2

Cancel the common factor.

$y = \frac{4 \cdot 5}{4 \cdot - 3}$

Step 3.2.4.3.1.2.3

Rewrite the expression.

$y = \frac{5}{- 3}$

Step 3.2.4.3.2

Move the negative in front of the fraction.

$y = - \frac{5}{3}$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{5}{3})$

Step 4

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - \frac{5}{3})$

Step 5