Precalculus Examples

$y = \frac{8 x^{2} - 16}{x - 4}$

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

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

$0 = \frac{8 x^{2} - 16}{x - 4}$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Set the numerator equal to zero.

$8 x^{2} - 16 = 0$

Step 1.2.2

Solve the equation for $x$ .

Tap for more steps...

Step 1.2.2.1

Add $16$ to both sides of the equation.

$8 x^{2} = 16$

Step 1.2.2.2

Divide each term in $8 x^{2} = 16$ by $8$ and simplify.

Tap for more steps...

Step 1.2.2.2.1

Divide each term in $8 x^{2} = 16$ by $8$ .

$\frac{8 x^{2}}{8} = \frac{16}{8}$

Step 1.2.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.2.2.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 1.2.2.2.2.1.1

Cancel the common factor.

$\frac{8 x^{2}}{8} = \frac{16}{8}$

Step 1.2.2.2.2.1.2

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

$x^{2} = \frac{16}{8}$

Step 1.2.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.2.2.3.1

Divide $16$ by $8$ .

$x^{2} = 2$

Step 1.2.2.3

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

$x = \pm \sqrt{2}$

Step 1.2.2.4

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

Tap for more steps...

Step 1.2.2.4.1

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

$x = \sqrt{2}$

Step 1.2.2.4.2

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

$x = - \sqrt{2}$

Step 1.2.2.4.3

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

$x = \sqrt{2}, - \sqrt{2}$

Step 1.3

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

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

$y = \frac{8 {(0)}^{2} - 16}{(0) - 4}$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = \frac{8 \cdot 0^{2} - 16}{(0) - 4}$

Step 2.2.2

Remove parentheses.

$y = \frac{8 \cdot 0^{2} - 16}{0 - 4}$

Step 2.2.3

Remove parentheses.

$y = \frac{8 {(0)}^{2} - 16}{(0) - 4}$

Step 2.2.4

Simplify $\frac{8 {(0)}^{2} - 16}{(0) - 4}$ .

Tap for more steps...

Step 2.2.4.1

Cancel the common factor of $8 {(0)}^{2} - 16$ and $(0) - 4$ .

Tap for more steps...

Step 2.2.4.1.1

Factor $4$ out of $8 {(0)}^{2}$ .

$y = \frac{4 (2 {(0)}^{2}) - 16}{(0) - 4}$

Step 2.2.4.1.2

Factor $4$ out of $- 16$ .

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

Step 2.2.4.1.3

Factor $4$ out of $4 (2 {(0)}^{2}) + 4 (- 4)$ .

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

Step 2.2.4.1.4

Cancel the common factors.

Tap for more steps...

Step 2.2.4.1.4.1

Factor $4$ out of $0$ .

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

Step 2.2.4.1.4.2

Factor $4$ out of $- 4$ .

$y = \frac{4 (2 {(0)}^{2} - 4)}{4 \cdot 0 + 4 \cdot - 1}$

Step 2.2.4.1.4.3

Factor $4$ out of $4 \cdot 0 + 4 \cdot - 1$ .

$y = \frac{4 (2 {(0)}^{2} - 4)}{4 \cdot (0 - 1)}$

Step 2.2.4.1.4.4

Cancel the common factor.

$y = \frac{4 (2 {(0)}^{2} - 4)}{4 \cdot (0 - 1)}$

Step 2.2.4.1.4.5

Rewrite the expression.

$y = \frac{2 {(0)}^{2} - 4}{0 - 1}$

Step 2.2.4.2

Simplify the numerator.

Tap for more steps...

Step 2.2.4.2.1

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

$y = \frac{2 \cdot 0 - 4}{0 - 1}$

Step 2.2.4.2.2

Multiply $2$ by $0$ .

$y = \frac{0 - 4}{0 - 1}$

Step 2.2.4.2.3

Subtract $4$ from $0$ .

$y = \frac{- 4}{0 - 1}$

Step 2.2.4.3

Simplify the expression.

Tap for more steps...

Step 2.2.4.3.1

Subtract $1$ from $0$ .

$y = \frac{- 4}{- 1}$

Step 2.2.4.3.2

Divide $- 4$ by $- 1$ .

$y = 4$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 4)$

Step 3

List the intersections.

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

y-intercept(s): $(0, 4)$

Step 4