Precalculus Examples

$f (x) = - 5 x^{4} + 20 x^{2}$

Step 1

Set $- 5 x^{4} + 20 x^{2}$ equal to $0$ .

$- 5 x^{4} + 20 x^{2} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

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

$- 5 {(x^{2})}^{2} + 20 x^{2} = 0$

Step 2.1.2

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

$- 5 u^{2} + 20 u = 0$

Step 2.1.3

Factor $5 u$ out of $- 5 u^{2} + 20 u$ .

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Step 2.1.3.1

Factor $5 u$ out of $- 5 u^{2}$ .

$5 u (- u) + 20 u = 0$

Step 2.1.3.2

Factor $5 u$ out of $20 u$ .

$5 u (- u) + 5 u (4) = 0$

Step 2.1.3.3

Factor $5 u$ out of $5 u (- u) + 5 u (4)$ .

$5 u (- u + 4) = 0$

Step 2.1.4

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

$5 x^{2} (- x^{2} + 4) = 0$

Step 2.2

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 2.3

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

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Step 2.3.1

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

$x^{2} = 0$

Step 2.3.2

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

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Step 2.3.2.1

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

$x = \pm \sqrt{0}$

Step 2.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

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

$x = \pm 0$

Step 2.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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Step 2.4.2.1

Subtract $4$ from both sides of the equation.

$- x^{2} = - 4$

Step 2.4.2.2

Divide each term in $- x^{2} = - 4$ by $- 1$ and simplify.

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Step 2.4.2.2.1

Divide each term in $- x^{2} = - 4$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 4}{- 1}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 4}{- 1}$

Step 2.4.2.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 4}{- 1}$

Step 2.4.2.2.3

Simplify the right side.

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Step 2.4.2.2.3.1

Divide $- 4$ by $- 1$ .

$x^{2} = 4$

Step 2.4.2.3

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

$x = \pm \sqrt{4}$

Step 2.4.2.4

Simplify $\pm \sqrt{4}$ .

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Step 2.4.2.4.1

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

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

Step 2.4.2.4.2

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

$x = \pm 2$

Step 2.4.2.5

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

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Step 2.4.2.5.1

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

$x = 2$

Step 2.4.2.5.2

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

$x = - 2$

Step 2.4.2.5.3

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

$x = 2, - 2$

Step 2.5

The final solution is all the values that make $5 x^{2} (- x^{2} + 4) = 0$ true.

$x = 0, 2, - 2$

Step 3