Trigonometry Examples

$2 x^{2} + 26 x + 156 = 0$

Step 1

Factor $2$ out of $2 x^{2} + 26 x + 156$ .

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Step 1.1

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

$2 (x^{2}) + 26 x + 156 = 0$

Step 1.2

Factor $2$ out of $26 x$ .

$2 (x^{2}) + 2 (13 x) + 156 = 0$

Step 1.3

Factor $2$ out of $156$ .

$2 x^{2} + 2 (13 x) + 2 \cdot 78 = 0$

Step 1.4

Factor $2$ out of $2 x^{2} + 2 (13 x)$ .

$2 (x^{2} + 13 x) + 2 \cdot 78 = 0$

Step 1.5

Factor $2$ out of $2 (x^{2} + 13 x) + 2 \cdot 78$ .

$2 (x^{2} + 13 x + 78) = 0$

Step 2

Divide each term in $2 (x^{2} + 13 x + 78) = 0$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 (x^{2} + 13 x + 78) = 0$ by $2$ .

$\frac{2 (x^{2} + 13 x + 78)}{2} = \frac{0}{2}$

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.

$\frac{2 (x^{2} + 13 x + 78)}{2} = \frac{0}{2}$

Step 2.2.1.2

Divide $x^{2} + 13 x + 78$ by $1$ .

$x^{2} + 13 x + 78 = \frac{0}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $2$ .

$x^{2} + 13 x + 78 = 0$

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 $a = 1$ , $b = 13$ , and $c = 78$ into the quadratic formula and solve for $x$ .

$\frac{- 13 \pm \sqrt{13^{2} - 4 \cdot (1 \cdot 78)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $13$ to the power of $2$ .

$x = \frac{- 13 \pm \sqrt{169 - 4 \cdot 1 \cdot 78}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 78$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 13 \pm \sqrt{169 - 4 \cdot 78}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $78$ .

$x = \frac{- 13 \pm \sqrt{169 - 312}}{2 \cdot 1}$

Step 5.1.3

Subtract $312$ from $169$ .

$x = \frac{- 13 \pm \sqrt{- 143}}{2 \cdot 1}$

Step 5.1.4

Rewrite $- 143$ as $- 1 (143)$ .

$x = \frac{- 13 \pm \sqrt{- 1 \cdot 143}}{2 \cdot 1}$

Step 5.1.5

Rewrite $\sqrt{- 1 (143)}$ as $\sqrt{- 1} \cdot \sqrt{143}$ .

$x = \frac{- 13 \pm \sqrt{- 1} \cdot \sqrt{143}}{2 \cdot 1}$

Step 5.1.6

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

$x = \frac{- 13 \pm i \sqrt{143}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{- 13 \pm i \sqrt{143}}{2}$

Step 6

The final answer is the combination of both solutions.

$x = - \frac{13 - i \sqrt{143}}{2}, - \frac{13 + i \sqrt{143}}{2}$