Algebra Examples

$y = x^{3} - 4 x$

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

$x$ .

$0 = x^{3} - 4 x$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as

$x^{3} - 4 x = 0$ .

$x^{3} - 4 x = 0$

Step 1.2.2

Factor the left side of the equation.

Tap for more steps...

Step 1.2.2.1

Factor

$x$ out of

$x^{3} - 4 x$ .

Tap for more steps...

Step 1.2.2.1.1

Factor

$x$ out of

$x^{3}$ .

$x \cdot x^{2} - 4 x = 0$

Step 1.2.2.1.2

Factor

$x$ out of

$- 4 x$ .

$x \cdot x^{2} + x \cdot - 4 = 0$

Step 1.2.2.1.3

Factor

$x$ out of

$x \cdot x^{2} + x \cdot - 4$ .

$x (x^{2} - 4) = 0$

$x (x^{2} - 4) = 0$

Step 1.2.2.2

Rewrite

$4$ as

$2^{2}$ .

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

Step 1.2.2.3

Factor.

Tap for more steps...

Step 1.2.2.3.1

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = x$ and

$b = 2$ .

$x ((x + 2) (x - 2)) = 0$

Step 1.2.2.3.2

Remove unnecessary parentheses.

$x (x + 2) (x - 2) = 0$

$x (x + 2) (x - 2) = 0$

$x (x + 2) (x - 2) = 0$

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

$0$ .

$x = 0$

$x + 2 = 0$

$x - 2 = 0$

Step 1.2.4

Set

$x$ equal to

$0$ .

$x = 0$

Step 1.2.5

Set

$x + 2$ equal to

$0$ and solve for

$x$ .

Tap for more steps...

Step 1.2.5.1

Set

$x + 2$ equal to

$0$ .

$x + 2 = 0$

Step 1.2.5.2

Subtract

$2$ from both sides of the equation.

$x = - 2$

$x = - 2$

Step 1.2.6

Set

$x - 2$ equal to

$0$ and solve for

$x$ .

Tap for more steps...

Step 1.2.6.1

Set

$x - 2$ equal to

$0$ .

$x - 2 = 0$

Step 1.2.6.2

Add

$2$ to both sides of the equation.

$x = 2$

$x = 2$

Step 1.2.7

The final solution is all the values that make

$x (x + 2) (x - 2) = 0$ true.

$x = 0, - 2, 2$

$x = 0, - 2, 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s):

$(0, 0), (- 2, 0), (2, 0)$

x-intercept(s):

$(0, 0), (- 2, 0), (2, 0)$

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

$y$ .

$y = {(0)}^{3} - 4 \cdot 0$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = 0^{3} - 4 \cdot 0$

Step 2.2.2

Remove parentheses.

$y = {(0)}^{3} - 4 \cdot 0$

Step 2.2.3

Simplify

${(0)}^{3} - 4 \cdot 0$ .

Tap for more steps...

Step 2.2.3.1

Simplify each term.

Tap for more steps...

Step 2.2.3.1.1

Raising

$0$ to any positive power yields

$0$ .

$y = 0 - 4 \cdot 0$

Step 2.2.3.1.2

Multiply

$- 4$ by

$0$ .

$y = 0 + 0$

$y = 0 + 0$

Step 2.2.3.2

Add

$0$ and

$0$ .

$y = 0$

$y = 0$

$y = 0$

Step 2.3

y-intercept(s) in point form.

y-intercept(s):

$(0, 0)$

y-intercept(s):

$(0, 0)$

Step 3

List the intersections.

x-intercept(s):

$(0, 0), (- 2, 0), (2, 0)$

y-intercept(s):

$(0, 0)$

Step 4

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$