Find the x and y Intercepts f(x)=(4x^2-16)/(5x^2+15x+10)

$f (x) = \frac{4 x^{2} - 16}{5 x^{2} + 15 x + 10}$

Step 1

Find the x-intercepts.

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

$0 = \frac{4 x^{2} - 16}{5 x^{2} + 15 x + 10}$

Solve the equation.

Set the numerator equal to zero.

$4 x^{2} - 16 = 0$

Solve the equation for $x$ .

Add $16$ to both sides of the equation.

$4 x^{2} = 16$

Divide each term in $4 x^{2} = 16$ by $4$ and simplify.

Divide each term in $4 x^{2} = 16$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{16}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{16}{4}$

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

$x^{2} = \frac{16}{4}$

Simplify the right side.

Divide $16$ by $4$ .

$x^{2} = 4$

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

$x = \pm \sqrt{4}$

Simplify $\pm \sqrt{4}$ .

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

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

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

$x = \pm 2$

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

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

$x = 2$

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

$x = - 2$

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

$x = 2, - 2$

Exclude the solutions that do not make $0 = \frac{4 x^{2} - 16}{5 x^{2} + 15 x + 10}$ true.

$x = 2$

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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

$y = \frac{4 {(0)}^{2} - 16}{5 {(0)}^{2} + 15 (0) + 10}$

Solve the equation.

Remove parentheses.

$y = \frac{4 \cdot 0^{2} - 16}{5 {(0)}^{2} + 15 (0) + 10}$

Remove parentheses.

$y = \frac{4 \cdot 0^{2} - 16}{5 \cdot 0^{2} + 15 (0) + 10}$

Remove parentheses.

$y = \frac{4 {(0)}^{2} - 16}{5 {(0)}^{2} + 15 (0) + 10}$

Simplify $\frac{4 {(0)}^{2} - 16}{5 {(0)}^{2} + 15 (0) + 10}$ .

Simplify the numerator.

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

$y = \frac{4 \cdot 0 - 16}{5 \cdot 0^{2} + 15 (0) + 10}$

Multiply $4$ by $0$ .

$y = \frac{0 - 16}{5 \cdot 0^{2} + 15 (0) + 10}$

Subtract $16$ from $0$ .

$y = \frac{- 16}{5 \cdot 0^{2} + 15 (0) + 10}$

Simplify the denominator.

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

$y = \frac{- 16}{5 \cdot 0 + 15 (0) + 10}$

Multiply $5$ by $0$ .

$y = \frac{- 16}{0 + 15 (0) + 10}$

Multiply $15$ by $0$ .

$y = \frac{- 16}{0 + 0 + 10}$

Add $0$ and $0$ .

$y = \frac{- 16}{0 + 10}$

Add $0$ and $10$ .

$y = \frac{- 16}{10}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 16$ and $10$ .

Factor $2$ out of $- 16$ .

$y = \frac{2 (- 8)}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$y = \frac{2 \cdot - 8}{2 \cdot 5}$

Cancel the common factor.

$y = \frac{2 \cdot - 8}{2 \cdot 5}$

Rewrite the expression.

$y = \frac{- 8}{5}$

Move the negative in front of the fraction.

$y = - \frac{8}{5}$

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{8}{5})$

Step 3

List the intersections.

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

y-intercept(s): $(0, - \frac{8}{5})$

Step 4

Find the x and y Intercepts f(x)=(4x^2-16)/(5x^2+15x+10)

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