Pre-Algebra Examples

$8 x^{2} - 2 x - 30 = 0$

Step 1

Factor $2$ out of $8 x^{2} - 2 x - 30$ .

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Step 1.1

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

$2 (4 x^{2}) - 2 x - 30 = 0$

Step 1.2

Factor $2$ out of $- 2 x$ .

$2 (4 x^{2}) + 2 (- x) - 30 = 0$

Step 1.3

Factor $2$ out of $- 30$ .

$2 (4 x^{2}) + 2 (- x) + 2 (- 15) = 0$

Step 1.4

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

$2 (4 x^{2} - x) + 2 (- 15) = 0$

Step 1.5

Factor $2$ out of $2 (4 x^{2} - x) + 2 (- 15)$ .

$2 (4 x^{2} - x - 15) = 0$

Step 2

Divide each term in $2 (4 x^{2} - x - 15) = 0$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 (4 x^{2} - x - 15) = 0$ by $2$ .

$\frac{2 (4 x^{2} - x - 15)}{2} = \frac{0}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 (4 x^{2} - x - 15)}{2} = \frac{0}{2}$

Step 2.2.1.2

Divide $4 x^{2} - x - 15$ by $1$ .

$4 x^{2} - x - 15 = \frac{0}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $2$ .

$4 x^{2} - x - 15 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 4$ , $b = - 1$ , and $c = - 15$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (4 \cdot - 15)}}{2 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 4 \cdot - 15}}{2 \cdot 4}$

Step 5.1.2

Multiply $- 4 \cdot 4 \cdot - 15$ .

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Step 5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{1 \pm \sqrt{1 - 16 \cdot - 15}}{2 \cdot 4}$

Step 5.1.2.2

Multiply $- 16$ by $- 15$ .

$x = \frac{1 \pm \sqrt{1 + 240}}{2 \cdot 4}$

Step 5.1.3

Add $1$ and $240$ .

$x = \frac{1 \pm \sqrt{241}}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$x = \frac{1 \pm \sqrt{241}}{8}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 4 \cdot - 15}}{2 \cdot 4}$

Step 6.1.2

Multiply $- 4 \cdot 4 \cdot - 15$ .

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Step 6.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{1 \pm \sqrt{1 - 16 \cdot - 15}}{2 \cdot 4}$

Step 6.1.2.2

Multiply $- 16$ by $- 15$ .

$x = \frac{1 \pm \sqrt{1 + 240}}{2 \cdot 4}$

Step 6.1.3

Add $1$ and $240$ .

$x = \frac{1 \pm \sqrt{241}}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$x = \frac{1 \pm \sqrt{241}}{8}$

Step 6.3

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{241}}{8}$

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

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 4 \cdot - 15}}{2 \cdot 4}$

Step 7.1.2

Multiply $- 4 \cdot 4 \cdot - 15$ .

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Step 7.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{1 \pm \sqrt{1 - 16 \cdot - 15}}{2 \cdot 4}$

Step 7.1.2.2

Multiply $- 16$ by $- 15$ .

$x = \frac{1 \pm \sqrt{1 + 240}}{2 \cdot 4}$

Step 7.1.3

Add $1$ and $240$ .

$x = \frac{1 \pm \sqrt{241}}{2 \cdot 4}$

Step 7.2

Multiply $2$ by $4$ .

$x = \frac{1 \pm \sqrt{241}}{8}$

Step 7.3

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{241}}{8}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{241}}{8}, \frac{1 - \sqrt{241}}{8}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{241}}{8}, \frac{1 - \sqrt{241}}{8}$

Decimal Form:

$x = 2.06552183 \dots, - 1.81552183 \dots$