Pre-Algebra Examples

$x^{2} - 8 x + 8 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 1$ , $b = - 8$ , and $c = 8$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot 8)}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4 \cdot 1 \cdot 8$ .

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Step 3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 8}}{2 \cdot 1}$

Step 3.1.2.2

Multiply $- 4$ by $8$ .

$x = \frac{8 \pm \sqrt{64 - 32}}{2 \cdot 1}$

Step 3.1.3

Subtract $32$ from $64$ .

$x = \frac{8 \pm \sqrt{32}}{2 \cdot 1}$

Step 3.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 3.1.4.1

Factor $16$ out of $32$ .

$x = \frac{8 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 3.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{8 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Step 3.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 4 \sqrt{2}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 4 \sqrt{2}}{2}$

Step 3.3

Simplify $\frac{8 \pm 4 \sqrt{2}}{2}$ .

$x = 4 \pm 2 \sqrt{2}$

Step 4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot 8$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 8}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $8$ .

$x = \frac{8 \pm \sqrt{64 - 32}}{2 \cdot 1}$

Step 4.1.3

Subtract $32$ from $64$ .

$x = \frac{8 \pm \sqrt{32}}{2 \cdot 1}$

Step 4.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 4.1.4.1

Factor $16$ out of $32$ .

$x = \frac{8 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 4.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{8 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Step 4.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 4 \sqrt{2}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 4 \sqrt{2}}{2}$

Step 4.3

Simplify $\frac{8 \pm 4 \sqrt{2}}{2}$ .

$x = 4 \pm 2 \sqrt{2}$

Step 4.4

Change the $\pm$ to $+$ .

$x = 4 + 2 \sqrt{2}$

Step 5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 8$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 8}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $8$ .

$x = \frac{8 \pm \sqrt{64 - 32}}{2 \cdot 1}$

Step 5.1.3

Subtract $32$ from $64$ .

$x = \frac{8 \pm \sqrt{32}}{2 \cdot 1}$

Step 5.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 5.1.4.1

Factor $16$ out of $32$ .

$x = \frac{8 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 5.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{8 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 4 \sqrt{2}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 4 \sqrt{2}}{2}$

Step 5.3

Simplify $\frac{8 \pm 4 \sqrt{2}}{2}$ .

$x = 4 \pm 2 \sqrt{2}$

Step 5.4

Change the $\pm$ to $-$ .

$x = 4 - 2 \sqrt{2}$

Step 6

The final answer is the combination of both solutions.

$x = 4 + 2 \sqrt{2}, 4 - 2 \sqrt{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 4 + 2 \sqrt{2}, 4 - 2 \sqrt{2}$

Decimal Form:

$x = 6.82842712 \dots, 1.17157287 \dots$