Precalculus Examples

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

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

Step 1

Use the quadratic formula to find the solutions.

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

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

Step 2

Substitute the values

a = 6

$a = 6$ ,

b = 12

$b = 12$ , and

c = 7

$c = 7$ into the quadratic formula and solve for

x

$x$ .

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

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise

12

$12$ to the power of

2

$2$ .

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

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

Step 3.1.2

Multiply

- 4 \cdot 6 \cdot 7

$- 4 \cdot 6 \cdot 7$ .

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Step 3.1.2.1

Multiply

- 4

$- 4$ by

6

$6$ .

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

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

Step 3.1.2.2

Multiply

- 24

$- 24$ by

7

$7$ .

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

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

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

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

Step 3.1.3

Subtract

168

$168$ from

144

$144$ .

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

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

Step 3.1.4

Rewrite

- 24

$- 24$ as

- 1 (24)

$- 1 (24)$ .

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

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

Step 3.1.5

Rewrite

\sqrt{- 1 (24)}

$\sqrt{- 1 (24)}$ as

\sqrt{- 1} \cdot \sqrt{24}

$\sqrt{- 1} \cdot \sqrt{24}$ .

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

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

Step 3.1.6

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

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

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

Step 3.1.7

Rewrite

24

$24$ as

2^{2} \cdot 6

$2^{2} \cdot 6$ .

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Step 3.1.7.1

Factor

4

$4$ out of

24

$24$ .

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

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

Step 3.1.7.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

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

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

Step 3.1.8

Pull terms out from under the radical.

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

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

Step 3.1.9

Move

2

$2$ to the left of

i

$i$ .

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

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

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

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

Step 3.2

Multiply

2

$2$ by

6

$6$ .

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

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

Step 3.3

Simplify

\frac{- 12 \pm 2 i \sqrt{6}}{12}

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

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

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

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

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

Step 4

The final answer is the combination of both solutions.

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

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