Algebra Examples

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

Set the function equal to $0$ and solve for $x$ .

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Replace $f (x)$ with $y$ .

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

To find the roots of the equation, replace $y$ with $0$ and solve.

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

Rewrite the equation as $x^{2} - 6 x - 9 = 0$ .

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

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Simplify the numerator.

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Raise $- 6$ to the power of $2$ to get $36$ .

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

Multiply $- 9$ by $1$ to get $- 9$ .

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

Multiply $- 4$ by $- 9$ to get $36$ .

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

Add $36$ and $36$ to get $72$ .

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

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

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

Pull terms out from under the radical.

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

Simplify the denominator.

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Rewrite.

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

Multiply $2$ by $1$ to get $2$ .

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

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

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

The final answer is the combination of both solutions.

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

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

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