Precalculus Examples

$6 x^{2} + 12 x + 7 = 0$

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 $a = 6$ , $b = 12$ , and $c = 7$ into the quadratic formula and solve for $x$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (6 \cdot 7)}}{2 \cdot 6}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 6 \cdot 7}}{2 \cdot 6}$

Step 3.1.2

Multiply $- 4 \cdot 6 \cdot 7$ .

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Step 3.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{- 12 \pm \sqrt{144 - 24 \cdot 7}}{2 \cdot 6}$

Step 3.1.2.2

Multiply $- 24$ by $7$ .

$x = \frac{- 12 \pm \sqrt{144 - 168}}{2 \cdot 6}$

Step 3.1.3

Subtract $168$ from $144$ .

$x = \frac{- 12 \pm \sqrt{- 24}}{2 \cdot 6}$

Step 3.1.4

Rewrite $- 24$ as $- 1 (24)$ .

$x = \frac{- 12 \pm \sqrt{- 1 \cdot 24}}{2 \cdot 6}$

Step 3.1.5

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$x = \frac{- 12 \pm \sqrt{- 1} \cdot \sqrt{24}}{2 \cdot 6}$

Step 3.1.6

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

$x = \frac{- 12 \pm i \cdot \sqrt{24}}{2 \cdot 6}$

Step 3.1.7

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 3.1.7.1

Factor $4$ out of $24$ .

$x = \frac{- 12 \pm i \cdot \sqrt{4 (6)}}{2 \cdot 6}$

Step 3.1.7.2

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

$x = \frac{- 12 \pm i \cdot \sqrt{2^{2} \cdot 6}}{2 \cdot 6}$

Step 3.1.8

Pull terms out from under the radical.

$x = \frac{- 12 \pm i \cdot (2 \sqrt{6})}{2 \cdot 6}$

Step 3.1.9

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

$x = \frac{- 12 \pm 2 i \sqrt{6}}{2 \cdot 6}$

Step 3.2

Multiply $2$ by $6$ .

$x = \frac{- 12 \pm 2 i \sqrt{6}}{12}$

Step 3.3

Simplify $\frac{- 12 \pm 2 i \sqrt{6}}{12}$ .

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

Step 4

The final answer is the combination of both solutions.

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