Precalculus Examples

$x^{3} - 3 x^{2} - 9 x = 0$

Step 1

Factor $x$ out of $x^{3} - 3 x^{2} - 9 x$ .

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Step 1.1

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

$x \cdot x^{2} - 3 x^{2} - 9 x = 0$

Step 1.2

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

$x \cdot x^{2} + x (- 3 x) - 9 x = 0$

Step 1.3

Factor $x$ out of $- 9 x$ .

$x \cdot x^{2} + x (- 3 x) + x \cdot - 9 = 0$

Step 1.4

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

$x (x^{2} - 3 x) + x \cdot - 9 = 0$

Step 1.5

Factor $x$ out of $x (x^{2} - 3 x) + x \cdot - 9$ .

$x (x^{2} - 3 x - 9) = 0$

Step 2

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

$x = 0$

$x^{2} - 3 x - 9 = 0$

Step 3

Set $x$ equal to $0$ .

$x = 0$

Step 4

Set $x^{2} - 3 x - 9$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{2} - 3 x - 9$ equal to $0$ .

$x^{2} - 3 x - 9 = 0$

Step 4.2

Solve $x^{2} - 3 x - 9 = 0$ for $x$ .

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Step 4.2.1

Use the quadratic formula to find the solutions.

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

Step 4.2.2

Substitute the values $a = 1$ , $b = - 3$ , and $c = - 9$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot - 9)}}{2 \cdot 1}$

Step 4.2.3

Simplify.

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Step 4.2.3.1

Simplify the numerator.

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Step 4.2.3.1.1

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

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

Step 4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 9$ .

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Step 4.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 4.2.3.1.2.2

Multiply $- 4$ by $- 9$ .

$x = \frac{3 \pm \sqrt{9 + 36}}{2 \cdot 1}$

Step 4.2.3.1.3

Add $9$ and $36$ .

$x = \frac{3 \pm \sqrt{45}}{2 \cdot 1}$

Step 4.2.3.1.4

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 4.2.3.1.4.1

Factor $9$ out of $45$ .

$x = \frac{3 \pm \sqrt{9 (5)}}{2 \cdot 1}$

Step 4.2.3.1.4.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{3 \pm \sqrt{3^{2} \cdot 5}}{2 \cdot 1}$

Step 4.2.3.1.5

Pull terms out from under the radical.

$x = \frac{3 \pm 3 \sqrt{5}}{2 \cdot 1}$

Step 4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm 3 \sqrt{5}}{2}$

Step 4.2.4

The final answer is the combination of both solutions.

$x = \frac{3 + 3 \sqrt{5}}{2}, \frac{3 - 3 \sqrt{5}}{2}$

Step 5

The final solution is all the values that make $x (x^{2} - 3 x - 9) = 0$ true.

$x = 0, \frac{3 + 3 \sqrt{5}}{2}, \frac{3 - 3 \sqrt{5}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{3 + 3 \sqrt{5}}{2}, \frac{3 - 3 \sqrt{5}}{2}$

Decimal Form:

$x = 0, 4.85410196 \dots, - 1.85410196 \dots$