Pre-Algebra Examples

$- 6 x^{2} - 4 x - 5 = 0$

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 $a = - 6$ , $b = - 4$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 6 \cdot - 5}}{2 \cdot - 6}$

Step 3.1.2

Multiply $- 4 \cdot - 6 \cdot - 5$ .

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Step 3.1.2.1

Multiply $- 4$ by $- 6$ .

$x = \frac{4 \pm \sqrt{16 + 24 \cdot - 5}}{2 \cdot - 6}$

Step 3.1.2.2

Multiply $24$ by $- 5$ .

$x = \frac{4 \pm \sqrt{16 - 120}}{2 \cdot - 6}$

Step 3.1.3

Subtract $120$ from $16$ .

$x = \frac{4 \pm \sqrt{- 104}}{2 \cdot - 6}$

Step 3.1.4

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

$x = \frac{4 \pm \sqrt{- 1 \cdot 104}}{2 \cdot - 6}$

Step 3.1.5

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

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{104}}{2 \cdot - 6}$

Step 3.1.6

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

$x = \frac{4 \pm i \cdot \sqrt{104}}{2 \cdot - 6}$

Step 3.1.7

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

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Step 3.1.7.1

Factor $4$ out of $104$ .

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

Step 3.1.7.2

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

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

Step 3.1.8

Pull terms out from under the radical.

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

Step 3.1.9

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

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

Step 3.2

Multiply $2$ by $- 6$ .

$x = \frac{4 \pm 2 i \sqrt{26}}{- 12}$

Step 3.3

Simplify $\frac{4 \pm 2 i \sqrt{26}}{- 12}$ .

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

Step 3.4

Move the negative in front of the fraction.

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

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 $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 6 \cdot - 5}}{2 \cdot - 6}$

Step 4.1.2

Multiply $- 4 \cdot - 6 \cdot - 5$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 6$ .

$x = \frac{4 \pm \sqrt{16 + 24 \cdot - 5}}{2 \cdot - 6}$

Step 4.1.2.2

Multiply $24$ by $- 5$ .

$x = \frac{4 \pm \sqrt{16 - 120}}{2 \cdot - 6}$

Step 4.1.3

Subtract $120$ from $16$ .

$x = \frac{4 \pm \sqrt{- 104}}{2 \cdot - 6}$

Step 4.1.4

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

$x = \frac{4 \pm \sqrt{- 1 \cdot 104}}{2 \cdot - 6}$

Step 4.1.5

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

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{104}}{2 \cdot - 6}$

Step 4.1.6

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

$x = \frac{4 \pm i \cdot \sqrt{104}}{2 \cdot - 6}$

Step 4.1.7

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

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Step 4.1.7.1

Factor $4$ out of $104$ .

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

Step 4.1.7.2

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

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

Step 4.1.8

Pull terms out from under the radical.

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

Step 4.1.9

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

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

Step 4.2

Multiply $2$ by $- 6$ .

$x = \frac{4 \pm 2 i \sqrt{26}}{- 12}$

Step 4.3

Simplify $\frac{4 \pm 2 i \sqrt{26}}{- 12}$ .

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

Step 4.4

Move the negative in front of the fraction.

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

Step 4.5

Change the $\pm$ to $+$ .

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

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 $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 6 \cdot - 5}}{2 \cdot - 6}$

Step 5.1.2

Multiply $- 4 \cdot - 6 \cdot - 5$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 6$ .

$x = \frac{4 \pm \sqrt{16 + 24 \cdot - 5}}{2 \cdot - 6}$

Step 5.1.2.2

Multiply $24$ by $- 5$ .

$x = \frac{4 \pm \sqrt{16 - 120}}{2 \cdot - 6}$

Step 5.1.3

Subtract $120$ from $16$ .

$x = \frac{4 \pm \sqrt{- 104}}{2 \cdot - 6}$

Step 5.1.4

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

$x = \frac{4 \pm \sqrt{- 1 \cdot 104}}{2 \cdot - 6}$

Step 5.1.5

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

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{104}}{2 \cdot - 6}$

Step 5.1.6

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

$x = \frac{4 \pm i \cdot \sqrt{104}}{2 \cdot - 6}$

Step 5.1.7

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

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Step 5.1.7.1

Factor $4$ out of $104$ .

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

Step 5.1.7.2

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

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

Step 5.1.8

Pull terms out from under the radical.

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

Step 5.1.9

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

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

Step 5.2

Multiply $2$ by $- 6$ .

$x = \frac{4 \pm 2 i \sqrt{26}}{- 12}$

Step 5.3

Simplify $\frac{4 \pm 2 i \sqrt{26}}{- 12}$ .

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

Step 5.4

Move the negative in front of the fraction.

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

Step 5.5

Change the $\pm$ to $-$ .

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

Step 6

The final answer is the combination of both solutions.

$x = - \frac{2 + i \sqrt{26}}{6}, - \frac{2 - i \sqrt{26}}{6}$