Examples

$f (x) = x^{2} - 6 x - 9$

Step 1

Set $x^{2} - 6 x - 9$ equal to $0$ .

$x^{2} - 6 x - 9 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Use the quadratic formula to find the solutions.

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

Step 2.2

Substitute the values $a = 1$ , $b = - 6$ , and $c = - 9$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot - 9)}}{2 \cdot 1}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 9}}{2 \cdot 1}$

Step 2.3.1.2.2

Multiply $- 4$ by $- 9$ .

$x = \frac{6 \pm \sqrt{36 + 36}}{2 \cdot 1}$

Step 2.3.1.3

Add $36$ and $36$ .

$x = \frac{6 \pm \sqrt{72}}{2 \cdot 1}$

Step 2.3.1.4

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 2.3.1.4.1

Factor $36$ out of $72$ .

$x = \frac{6 \pm \sqrt{36 (2)}}{2 \cdot 1}$

Step 2.3.1.4.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{6 \pm \sqrt{6^{2} \cdot 2}}{2 \cdot 1}$

Step 2.3.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 6 \sqrt{2}}{2 \cdot 1}$

Step 2.3.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 6 \sqrt{2}}{2}$

Step 2.3.3

Simplify $\frac{6 \pm 6 \sqrt{2}}{2}$ .

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

Step 2.4

The final answer is the combination of both solutions.

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

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 7.24264068 \dots, - 1.24264068 \dots$

Step 4

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