Calculus Examples

$15 x^{4} - 60 x^{2} = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

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

$15 {(x^{2})}^{2} - 60 x^{2} = 0$

Step 1.2

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

$15 u^{2} - 60 u = 0$

Step 1.3

Factor $15 u$ out of $15 u^{2} - 60 u$ .

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Step 1.3.1

Factor $15 u$ out of $15 u^{2}$ .

$15 u (u) - 60 u = 0$

Step 1.3.2

Factor $15 u$ out of $- 60 u$ .

$15 u (u) + 15 u (- 4) = 0$

Step 1.3.3

Factor $15 u$ out of $15 u (u) + 15 u (- 4)$ .

$15 u (u - 4) = 0$

Step 1.4

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

$15 x^{2} (x^{2} - 4) = 0$

Step 2

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

$x^{2} = 0$

$x^{2} - 4 = 0$

Step 3

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

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Step 3.1

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

$x^{2} = 0$

Step 3.2

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

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Step 3.2.1

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

$x = \pm \sqrt{0}$

Step 3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 3.2.2.1

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

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

Step 3.2.2.2

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

$x = \pm 0$

Step 3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

Set $x^{2} - 4$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{2} - 4$ equal to $0$ .

$x^{2} - 4 = 0$

Step 4.2

Solve $x^{2} - 4 = 0$ for $x$ .

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Step 4.2.1

Add $4$ to both sides of the equation.

$x^{2} = 4$

Step 4.2.2

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

$x = \pm \sqrt{4}$

Step 4.2.3

Simplify $\pm \sqrt{4}$ .

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Step 4.2.3.1

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

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

Step 4.2.3.2

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

$x = \pm 2$

Step 4.2.4

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

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Step 4.2.4.1

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

$x = 2$

Step 4.2.4.2

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

$x = - 2$

Step 4.2.4.3

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

$x = 2, - 2$

Step 5

The final solution is all the values that make $15 x^{2} (x^{2} - 4) = 0$ true.

$x = 0, 2, - 2$