Pre-Algebra Examples

$x (x + 6) = 8 x^{2} + 5$

Step 1

Simplify $x (x + 6)$ .

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Step 1.1

Rewrite.

$0 + 0 + x (x + 6) = 8 x^{2} + 5$

Step 1.2

Simplify by adding zeros.

$x (x + 6) = 8 x^{2} + 5$

Step 1.3

Apply the distributive property.

$x \cdot x + x \cdot 6 = 8 x^{2} + 5$

Step 1.4

Simplify the expression.

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Step 1.4.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 6 = 8 x^{2} + 5$

Step 1.4.2

Move $6$ to the left of $x$ .

$x^{2} + 6 x = 8 x^{2} + 5$

Step 2

Move all terms containing $x$ to the left side of the equation.

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Step 2.1

Subtract $8 x^{2}$ from both sides of the equation.

$x^{2} + 6 x - 8 x^{2} = 5$

Step 2.2

Subtract $8 x^{2}$ from $x^{2}$ .

$- 7 x^{2} + 6 x = 5$

Step 3

Subtract $5$ from both sides of the equation.

$- 7 x^{2} + 6 x - 5 = 0$

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 $a = - 7$ , $b = 6$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (- 7 \cdot - 5)}}{2 \cdot - 7}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 7 \cdot - 5}}{2 \cdot - 7}$

Step 6.1.2

Multiply $- 4 \cdot - 7 \cdot - 5$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 7$ .

$x = \frac{- 6 \pm \sqrt{36 + 28 \cdot - 5}}{2 \cdot - 7}$

Step 6.1.2.2

Multiply $28$ by $- 5$ .

$x = \frac{- 6 \pm \sqrt{36 - 140}}{2 \cdot - 7}$

Step 6.1.3

Subtract $140$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 104}}{2 \cdot - 7}$

Step 6.1.4

Rewrite $- 104$ as $- 1 (104)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 104}}{2 \cdot - 7}$

Step 6.1.5

Rewrite $\sqrt{- 1 (104)}$ as $\sqrt{- 1} \cdot \sqrt{104}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{104}}{2 \cdot - 7}$

Step 6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 6 \pm i \cdot \sqrt{104}}{2 \cdot - 7}$

Step 6.1.7

Rewrite $104$ as $2^{2} \cdot 26$ .

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Step 6.1.7.1

Factor $4$ out of $104$ .

$x = \frac{- 6 \pm i \cdot \sqrt{4 (26)}}{2 \cdot - 7}$

Step 6.1.7.2

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

$x = \frac{- 6 \pm i \cdot \sqrt{2^{2} \cdot 26}}{2 \cdot - 7}$

Step 6.1.8

Pull terms out from under the radical.

$x = \frac{- 6 \pm i \cdot (2 \sqrt{26})}{2 \cdot - 7}$

Step 6.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 6 \pm 2 i \sqrt{26}}{2 \cdot - 7}$

Step 6.2

Multiply $2$ by $- 7$ .

$x = \frac{- 6 \pm 2 i \sqrt{26}}{- 14}$

Step 6.3

Simplify $\frac{- 6 \pm 2 i \sqrt{26}}{- 14}$ .

$x = \frac{3 \pm i \sqrt{26}}{7}$

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 $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 7 \cdot - 5}}{2 \cdot - 7}$

Step 7.1.2

Multiply $- 4 \cdot - 7 \cdot - 5$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 7$ .

$x = \frac{- 6 \pm \sqrt{36 + 28 \cdot - 5}}{2 \cdot - 7}$

Step 7.1.2.2

Multiply $28$ by $- 5$ .

$x = \frac{- 6 \pm \sqrt{36 - 140}}{2 \cdot - 7}$

Step 7.1.3

Subtract $140$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 104}}{2 \cdot - 7}$

Step 7.1.4

Rewrite $- 104$ as $- 1 (104)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 104}}{2 \cdot - 7}$

Step 7.1.5

Rewrite $\sqrt{- 1 (104)}$ as $\sqrt{- 1} \cdot \sqrt{104}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{104}}{2 \cdot - 7}$

Step 7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 6 \pm i \cdot \sqrt{104}}{2 \cdot - 7}$

Step 7.1.7

Rewrite $104$ as $2^{2} \cdot 26$ .

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Step 7.1.7.1

Factor $4$ out of $104$ .

$x = \frac{- 6 \pm i \cdot \sqrt{4 (26)}}{2 \cdot - 7}$

Step 7.1.7.2

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

$x = \frac{- 6 \pm i \cdot \sqrt{2^{2} \cdot 26}}{2 \cdot - 7}$

Step 7.1.8

Pull terms out from under the radical.

$x = \frac{- 6 \pm i \cdot (2 \sqrt{26})}{2 \cdot - 7}$

Step 7.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 6 \pm 2 i \sqrt{26}}{2 \cdot - 7}$

Step 7.2

Multiply $2$ by $- 7$ .

$x = \frac{- 6 \pm 2 i \sqrt{26}}{- 14}$

Step 7.3

Simplify $\frac{- 6 \pm 2 i \sqrt{26}}{- 14}$ .

$x = \frac{3 \pm i \sqrt{26}}{7}$

Step 7.4

Change the $\pm$ to $+$ .

$x = \frac{3 + i \sqrt{26}}{7}$

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

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

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 7 \cdot - 5}}{2 \cdot - 7}$

Step 8.1.2

Multiply $- 4 \cdot - 7 \cdot - 5$ .

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Step 8.1.2.1

Multiply $- 4$ by $- 7$ .

$x = \frac{- 6 \pm \sqrt{36 + 28 \cdot - 5}}{2 \cdot - 7}$

Step 8.1.2.2

Multiply $28$ by $- 5$ .

$x = \frac{- 6 \pm \sqrt{36 - 140}}{2 \cdot - 7}$

Step 8.1.3

Subtract $140$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 104}}{2 \cdot - 7}$

Step 8.1.4

Rewrite $- 104$ as $- 1 (104)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 104}}{2 \cdot - 7}$

Step 8.1.5

Rewrite $\sqrt{- 1 (104)}$ as $\sqrt{- 1} \cdot \sqrt{104}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{104}}{2 \cdot - 7}$

Step 8.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 6 \pm i \cdot \sqrt{104}}{2 \cdot - 7}$

Step 8.1.7

Rewrite $104$ as $2^{2} \cdot 26$ .

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Step 8.1.7.1

Factor $4$ out of $104$ .

$x = \frac{- 6 \pm i \cdot \sqrt{4 (26)}}{2 \cdot - 7}$

Step 8.1.7.2

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

$x = \frac{- 6 \pm i \cdot \sqrt{2^{2} \cdot 26}}{2 \cdot - 7}$

Step 8.1.8

Pull terms out from under the radical.

$x = \frac{- 6 \pm i \cdot (2 \sqrt{26})}{2 \cdot - 7}$

Step 8.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 6 \pm 2 i \sqrt{26}}{2 \cdot - 7}$

Step 8.2

Multiply $2$ by $- 7$ .

$x = \frac{- 6 \pm 2 i \sqrt{26}}{- 14}$

Step 8.3

Simplify $\frac{- 6 \pm 2 i \sqrt{26}}{- 14}$ .

$x = \frac{3 \pm i \sqrt{26}}{7}$

Step 8.4

Change the $\pm$ to $-$ .

$x = \frac{3 - i \sqrt{26}}{7}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{3 + i \sqrt{26}}{7}, \frac{3 - i \sqrt{26}}{7}$