Find the x and y Intercepts f(x)=(4x^2-9)/(2x-4)

$f (x) = \frac{4 x^{2} - 9}{2 x - 4}$

Step 1

Find the x-intercepts.

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

$0 = \frac{4 x^{2} - 9}{2 x - 4}$

Solve the equation.

Set the numerator equal to zero.

$4 x^{2} - 9 = 0$

Solve the equation for $x$ .

Add $9$ to both sides of the equation.

$4 x^{2} = 9$

Divide each term in $4 x^{2} = 9$ by $4$ and simplify.

Divide each term in $4 x^{2} = 9$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{9}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{9}{4}$

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

$x^{2} = \frac{9}{4}$

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

$x = \pm \sqrt{\frac{9}{4}}$

Simplify $\pm \sqrt{\frac{9}{4}}$ .

Rewrite $\sqrt{\frac{9}{4}}$ as $\frac{\sqrt{9}}{\sqrt{4}}$ .

$x = \pm \frac{\sqrt{9}}{\sqrt{4}}$

Simplify the numerator.

Rewrite $9$ as $3^{2}$ .

$x = \pm \frac{\sqrt{3^{2}}}{\sqrt{4}}$

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

$x = \pm \frac{3}{\sqrt{4}}$

Simplify the denominator.

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

$x = \pm \frac{3}{\sqrt{2^{2}}}$

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

$x = \pm \frac{3}{2}$

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.

$x = \frac{3}{2}$

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

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

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

$x = \frac{3}{2}, - \frac{3}{2}$

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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

$y = \frac{4 {(0)}^{2} - 9}{2 (0) - 4}$

Solve the equation.

Remove parentheses.

$y = \frac{4 \cdot 0^{2} - 9}{2 (0) - 4}$

Remove parentheses.

$y = \frac{4 {(0)}^{2} - 9}{2 (0) - 4}$

Simplify $\frac{4 {(0)}^{2} - 9}{2 (0) - 4}$ .

Simplify the numerator.

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

$y = \frac{4 \cdot 0 - 9}{2 (0) - 4}$

Multiply $4$ by $0$ .

$y = \frac{0 - 9}{2 (0) - 4}$

Subtract $9$ from $0$ .

$y = \frac{- 9}{2 (0) - 4}$

Simplify the denominator.

Multiply $2$ by $0$ .

$y = \frac{- 9}{0 - 4}$

Subtract $4$ from $0$ .

$y = \frac{- 9}{- 4}$

Dividing two negative values results in a positive value.

$y = \frac{9}{4}$

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{9}{4})$

Step 3

List the intersections.

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

y-intercept(s): $(0, \frac{9}{4})$

Step 4

Find the x and y Intercepts f(x)=(4x^2-9)/(2x-4)

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