Calculus Examples

$y^{2} = x^{3} - 4 x$

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 $x$ .

${(0)}^{2} = x^{3} - 4 x$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $x^{3} - 4 x = {(0)}^{2}$ .

$x^{3} - 4 x = {(0)}^{2}$

Step 1.2.2

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

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

Step 1.2.3

Factor the left side of the equation.

Tap for more steps...

Step 1.2.3.1

Factor $x$ out of $x^{3} - 4 x$ .

Tap for more steps...

Step 1.2.3.1.1

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

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

Step 1.2.3.1.2

Factor $x$ out of $- 4 x$ .

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

Step 1.2.3.1.3

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

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

Step 1.2.3.2

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

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

Step 1.2.3.3

Factor.

Tap for more steps...

Step 1.2.3.3.1

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 1.2.3.3.2

Remove unnecessary parentheses.

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

Step 1.2.4

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 1.2.5

Set $x$ equal to $0$ .

$x = 0$

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

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.7

Set $x - 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.7.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 1.2.7.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.8

The final solution is all the values that make $x (x + 2) (x - 2) = 0$ true.

$x = 0, - 2, 2$

Step 1.3

x-intercept(s) in point form.

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 $y$ .

$y^{2} = {(0)}^{3} - 4 \cdot 0$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Multiply $- 4$ by $0$ .

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

Step 2.2.2

Add ${(0)}^{3}$ and $0$ .

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

Step 2.2.3

Remove parentheses.

$y^{2} = 0^{3}$

Step 2.2.4

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

$y = \pm \sqrt{0^{3}}$

Step 2.2.5

Simplify $\pm \sqrt{0^{3}}$ .

Tap for more steps...

Step 2.2.5.1

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

$y = \pm \sqrt{0}$

Step 2.2.5.2

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

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

Step 2.2.5.3

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

$y = \pm 0$

Step 2.2.5.4

Plus or minus $0$ is $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (- 2, 0), (2, 0)$

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

Step 4