Precalculus Examples

$f (x) = x^{4} - 3 x^{3} - 20 x^{2}$

Step 1

Set $x^{4} - 3 x^{3} - 20 x^{2}$ equal to $0$ .

$x^{4} - 3 x^{3} - 20 x^{2} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $x^{2}$ out of $x^{4} - 3 x^{3} - 20 x^{2}$ .

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Step 2.1.1

Factor $x^{2}$ out of $x^{4}$ .

$x^{2} x^{2} - 3 x^{3} - 20 x^{2} = 0$

Step 2.1.2

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

$x^{2} x^{2} + x^{2} (- 3 x) - 20 x^{2} = 0$

Step 2.1.3

Factor $x^{2}$ out of $- 20 x^{2}$ .

$x^{2} x^{2} + x^{2} (- 3 x) + x^{2} \cdot - 20 = 0$

Step 2.1.4

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

$x^{2} (x^{2} - 3 x) + x^{2} \cdot - 20 = 0$

Step 2.1.5

Factor $x^{2}$ out of $x^{2} (x^{2} - 3 x) + x^{2} \cdot - 20$ .

$x^{2} (x^{2} - 3 x - 20) = 0$

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 $0$ .

$x^{2} = 0$

$x^{2} - 3 x - 20 = 0$

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 $x^{2} - 3 x - 20$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{2} - 3 x - 20$ equal to $0$ .

$x^{2} - 3 x - 20 = 0$

Step 2.4.2

Solve $x^{2} - 3 x - 20 = 0$ for $x$ .

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Step 2.4.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.4.2.2

Substitute the values $a = 1$ , $b = - 3$ , and $c = - 20$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot - 20)}}{2 \cdot 1}$

Step 2.4.2.3

Simplify.

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Step 2.4.2.3.1

Simplify the numerator.

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Step 2.4.2.3.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 20}}{2 \cdot 1}$

Step 2.4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 20$ .

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Step 2.4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 20}}{2 \cdot 1}$

Step 2.4.2.3.1.2.2

Multiply $- 4$ by $- 20$ .

$x = \frac{3 \pm \sqrt{9 + 80}}{2 \cdot 1}$

Step 2.4.2.3.1.3

Add $9$ and $80$ .

$x = \frac{3 \pm \sqrt{89}}{2 \cdot 1}$

Step 2.4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{89}}{2}$

Step 2.4.2.4

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{89}}{2}, \frac{3 - \sqrt{89}}{2}$

Step 2.5

The final solution is all the values that make $x^{2} (x^{2} - 3 x - 20) = 0$ true.

$x = 0, \frac{3 + \sqrt{89}}{2}, \frac{3 - \sqrt{89}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{3 + \sqrt{89}}{2}, \frac{3 - \sqrt{89}}{2}$

Decimal Form:

$x = 0, 6.21699056 \dots, - 3.21699056 \dots$

Step 4