Precalculus Examples

$y = x^{3} - 3 x^{2}$

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 $x$ .

$0 = x^{3} - 3 x^{2}$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $x^{3} - 3 x^{2} = 0$ .

$x^{3} - 3 x^{2} = 0$

Step 1.2.2

Factor $x^{2}$ out of $x^{3} - 3 x^{2}$ .

Tap for more steps...

Step 1.2.2.1

Factor $x^{2}$ out of $x^{3}$ .

$x^{2} x - 3 x^{2} = 0$

Step 1.2.2.2

Factor $x^{2}$ out of $- 3 x^{2}$ .

$x^{2} x + x^{2} \cdot - 3 = 0$

Step 1.2.2.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot - 3$ .

$x^{2} (x - 3) = 0$

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 $0$ .

$x^{2} = 0$

$x - 3 = 0$

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 $x - 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.5.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 1.2.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 1.2.6

The final solution is all the values that make $x^{2} (x - 3) = 0$ true.

$x = 0, 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (3, 0)$

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 $y$ .

$y = {(0)}^{3} - 3 {(0)}^{2}$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = 0^{3} - 3 {(0)}^{2}$

Step 2.2.2

Remove parentheses.

$y = 0^{3} - 3 \cdot 0^{2}$

Step 2.2.3

Remove parentheses.

$y = {(0)}^{3} - 3 {(0)}^{2}$

Step 2.2.4

Simplify ${(0)}^{3} - 3 {(0)}^{2}$ .

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 $0$ .

$y = 0 - 3 {(0)}^{2}$

Step 2.2.4.1.2

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

$y = 0 - 3 \cdot 0$

Step 2.2.4.1.3

Multiply $- 3$ 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): $(0, 0), (3, 0)$

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

Step 4