Algebra Examples

$f (x) = x^{4} - 16 x^{2}$

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 $x$ .

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

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $x^{4} - 16 x^{2} = 0$ .

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

Step 1.2.2

Factor the left side of the equation.

Tap for more steps...

Step 1.2.2.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

${(x^{2})}^{2} - 16 x^{2} = 0$

Step 1.2.2.2

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$u^{2} - 16 u = 0$

Step 1.2.2.3

Factor $u$ out of $u^{2} - 16 u$ .

Tap for more steps...

Step 1.2.2.3.1

Factor $u$ out of $u^{2}$ .

$u \cdot u - 16 u = 0$

Step 1.2.2.3.2

Factor $u$ out of $- 16 u$ .

$u \cdot u + u \cdot - 16 = 0$

Step 1.2.2.3.3

Factor $u$ out of $u \cdot u + u \cdot - 16$ .

$u (u - 16) = 0$

Step 1.2.2.4

Replace all occurrences of $u$ with $x^{2}$ .

$x^{2} (x^{2} - 16) = 0$

Step 1.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

$x^{2} - 16 = 0$

Step 1.2.4

Set $x^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.4.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 1.2.4.2

Solve $x^{2} = 0$ for $x$ .

Tap for more steps...

Step 1.2.4.2.1

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

$x = \pm \sqrt{0}$

Step 1.2.4.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 1.2.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.2.4.2.2.2

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

$x = \pm 0$

Step 1.2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.2.5

Set $x^{2} - 16$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.5.1

Set $x^{2} - 16$ equal to $0$ .

$x^{2} - 16 = 0$

Step 1.2.5.2

Solve $x^{2} - 16 = 0$ for $x$ .

Tap for more steps...

Step 1.2.5.2.1

Add $16$ to both sides of the equation.

$x^{2} = 16$

Step 1.2.5.2.2

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

$x = \pm \sqrt{16}$

Step 1.2.5.2.3

Simplify $\pm \sqrt{16}$ .

Tap for more steps...

Step 1.2.5.2.3.1

Rewrite $16$ as $4^{2}$ .

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

Step 1.2.5.2.3.2

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

$x = \pm 4$

Step 1.2.5.2.4

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

Tap for more steps...

Step 1.2.5.2.4.1

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

$x = 4$

Step 1.2.5.2.4.2

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

$x = - 4$

Step 1.2.5.2.4.3

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

$x = 4, - 4$

Step 1.2.6

The final solution is all the values that make $x^{2} (x^{2} - 16) = 0$ true.

$x = 0, 4, - 4$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

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

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Remove parentheses.

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

Step 2.2.4

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

Tap for more steps...

Step 2.2.4.1

Simplify each term.

Tap for more steps...

Step 2.2.4.1.1

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

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

Step 2.2.4.1.2

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

$y = 0 - 16 \cdot 0$

Step 2.2.4.1.3

Multiply $- 16$ by $0$ .

$y = 0 + 0$

Step 2.2.4.2

Add $0$ and $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4