Pre-Algebra Examples

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

Step 1

Subtract $8$ from both sides of the equation.

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

Step 2

Factor $2$ out of $8 x^{2} + 2 x - 8$ .

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Step 2.1

Factor $2$ out of $8 x^{2}$ .

$2 (4 x^{2}) + 2 x - 8 = 0$

Step 2.2

Factor $2$ out of $2 x$ .

$2 (4 x^{2}) + 2 (x) - 8 = 0$

Step 2.3

Factor $2$ out of $- 8$ .

$2 (4 x^{2}) + 2 x + 2 \cdot - 4 = 0$

Step 2.4

Factor $2$ out of $2 (4 x^{2}) + 2 x$ .

$2 (4 x^{2} + x) + 2 \cdot - 4 = 0$

Step 2.5

Factor $2$ out of $2 (4 x^{2} + x) + 2 \cdot - 4$ .

$2 (4 x^{2} + x - 4) = 0$

Step 3

Divide each term in $2 (4 x^{2} + x - 4) = 0$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 (4 x^{2} + x - 4) = 0$ by $2$ .

$\frac{2 (4 x^{2} + x - 4)}{2} = \frac{0}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 (4 x^{2} + x - 4)}{2} = \frac{0}{2}$

Step 3.2.1.2

Divide $4 x^{2} + x - 4$ by $1$ .

$4 x^{2} + x - 4 = \frac{0}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $2$ .

$4 x^{2} + x - 4 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 4$ , $b = 1$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

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

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

One to any power is one.

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

Step 6.1.2

Multiply $- 4 \cdot 4 \cdot - 4$ .

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Step 6.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 1 \pm \sqrt{1 - 16 \cdot - 4}}{2 \cdot 4}$

Step 6.1.2.2

Multiply $- 16$ by $- 4$ .

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

Step 6.1.3

Add $1$ and $64$ .

$x = \frac{- 1 \pm \sqrt{65}}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$x = \frac{- 1 \pm \sqrt{65}}{8}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

One to any power is one.

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

Step 7.1.2

Multiply $- 4 \cdot 4 \cdot - 4$ .

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Step 7.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 1 \pm \sqrt{1 - 16 \cdot - 4}}{2 \cdot 4}$

Step 7.1.2.2

Multiply $- 16$ by $- 4$ .

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

Step 7.1.3

Add $1$ and $64$ .

$x = \frac{- 1 \pm \sqrt{65}}{2 \cdot 4}$

Step 7.2

Multiply $2$ by $4$ .

$x = \frac{- 1 \pm \sqrt{65}}{8}$

Step 7.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{65}}{8}$

Step 7.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 + \sqrt{65}}{8}$

Step 7.5

Factor $- 1$ out of $\sqrt{65}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{65})}{8}$

Step 7.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{65})$ .

$x = \frac{- 1 (1 - \sqrt{65})}{8}$

Step 7.7

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{65}}{8}$

Step 8

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

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Step 8.1

Simplify the numerator.

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Step 8.1.1

One to any power is one.

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

Step 8.1.2

Multiply $- 4 \cdot 4 \cdot - 4$ .

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Step 8.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 1 \pm \sqrt{1 - 16 \cdot - 4}}{2 \cdot 4}$

Step 8.1.2.2

Multiply $- 16$ by $- 4$ .

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

Step 8.1.3

Add $1$ and $64$ .

$x = \frac{- 1 \pm \sqrt{65}}{2 \cdot 4}$

Step 8.2

Multiply $2$ by $4$ .

$x = \frac{- 1 \pm \sqrt{65}}{8}$

Step 8.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{65}}{8}$

Step 8.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 - \sqrt{65}}{8}$

Step 8.5

Factor $- 1$ out of $- \sqrt{65}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{65})}{8}$

Step 8.6

Factor $- 1$ out of $- 1 (1) - (\sqrt{65})$ .

$x = \frac{- 1 (1 + \sqrt{65})}{8}$

Step 8.7

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{65}}{8}$

Step 9

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{65}}{8}, - \frac{1 + \sqrt{65}}{8}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 - \sqrt{65}}{8}, - \frac{1 + \sqrt{65}}{8}$

Decimal Form:

$x = 0.88278221 \dots, - 1.13278221 \dots$