Trigonometry Examples

$g (x) = 3 x^{2} + 5 x + 11$ , $g (x) = - 5$

Step 1

Substitute $- 5$ for $g (x)$ .

$- 5 = 3 x^{2} + 5 x + 11$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $3 x^{2} + 5 x + 11 = - 5$ .

$3 x^{2} + 5 x + 11 = - 5$

Step 2.2

Move all terms to the left side of the equation and simplify.

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Step 2.2.1

Add $5$ to both sides of the equation.

$3 x^{2} + 5 x + 11 + 5 = 0$

Step 2.2.2

Add $11$ and $5$ .

$3 x^{2} + 5 x + 16 = 0$

Step 2.3

Use the quadratic formula to find the solutions.

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

Step 2.4

Substitute the values $a = 3$ , $b = 5$ , and $c = 16$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (3 \cdot 16)}}{2 \cdot 3}$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 3 \cdot 16}}{2 \cdot 3}$

Step 2.5.1.2

Multiply $- 4 \cdot 3 \cdot 16$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 5 \pm \sqrt{25 - 12 \cdot 16}}{2 \cdot 3}$

Step 2.5.1.2.2

Multiply $- 12$ by $16$ .

$x = \frac{- 5 \pm \sqrt{25 - 192}}{2 \cdot 3}$

Step 2.5.1.3

Subtract $192$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 167}}{2 \cdot 3}$

Step 2.5.1.4

Rewrite $- 167$ as $- 1 (167)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 167}}{2 \cdot 3}$

Step 2.5.1.5

Rewrite $\sqrt{- 1 (167)}$ as $\sqrt{- 1} \cdot \sqrt{167}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{167}}{2 \cdot 3}$

Step 2.5.1.6

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

$x = \frac{- 5 \pm i \sqrt{167}}{2 \cdot 3}$

Step 2.5.2

Multiply $2$ by $3$ .

$x = \frac{- 5 \pm i \sqrt{167}}{6}$

Step 2.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 3 \cdot 16}}{2 \cdot 3}$

Step 2.6.1.2

Multiply $- 4 \cdot 3 \cdot 16$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 5 \pm \sqrt{25 - 12 \cdot 16}}{2 \cdot 3}$

Step 2.6.1.2.2

Multiply $- 12$ by $16$ .

$x = \frac{- 5 \pm \sqrt{25 - 192}}{2 \cdot 3}$

Step 2.6.1.3

Subtract $192$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 167}}{2 \cdot 3}$

Step 2.6.1.4

Rewrite $- 167$ as $- 1 (167)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 167}}{2 \cdot 3}$

Step 2.6.1.5

Rewrite $\sqrt{- 1 (167)}$ as $\sqrt{- 1} \cdot \sqrt{167}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{167}}{2 \cdot 3}$

Step 2.6.1.6

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

$x = \frac{- 5 \pm i \sqrt{167}}{2 \cdot 3}$

Step 2.6.2

Multiply $2$ by $3$ .

$x = \frac{- 5 \pm i \sqrt{167}}{6}$

Step 2.6.3

Change the $\pm$ to $+$ .

$x = \frac{- 5 + i \sqrt{167}}{6}$

Step 2.6.4

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 + i \sqrt{167}}{6}$

Step 2.6.5

Factor $- 1$ out of $i \sqrt{167}$ .

$x = \frac{- 1 \cdot 5 - (- i \sqrt{167})}{6}$

Step 2.6.6

Factor $- 1$ out of $- 1 (5) - (- i \sqrt{167})$ .

$x = \frac{- 1 (5 - i \sqrt{167})}{6}$

Step 2.6.7

Move the negative in front of the fraction.

$x = - \frac{5 - i \sqrt{167}}{6}$

Step 2.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 3 \cdot 16}}{2 \cdot 3}$

Step 2.7.1.2

Multiply $- 4 \cdot 3 \cdot 16$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 5 \pm \sqrt{25 - 12 \cdot 16}}{2 \cdot 3}$

Step 2.7.1.2.2

Multiply $- 12$ by $16$ .

$x = \frac{- 5 \pm \sqrt{25 - 192}}{2 \cdot 3}$

Step 2.7.1.3

Subtract $192$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 167}}{2 \cdot 3}$

Step 2.7.1.4

Rewrite $- 167$ as $- 1 (167)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 167}}{2 \cdot 3}$

Step 2.7.1.5

Rewrite $\sqrt{- 1 (167)}$ as $\sqrt{- 1} \cdot \sqrt{167}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{167}}{2 \cdot 3}$

Step 2.7.1.6

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

$x = \frac{- 5 \pm i \sqrt{167}}{2 \cdot 3}$

Step 2.7.2

Multiply $2$ by $3$ .

$x = \frac{- 5 \pm i \sqrt{167}}{6}$

Step 2.7.3

Change the $\pm$ to $-$ .

$x = \frac{- 5 - i \sqrt{167}}{6}$

Step 2.7.4

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 - i \sqrt{167}}{6}$

Step 2.7.5

Factor $- 1$ out of $- i \sqrt{167}$ .

$x = \frac{- 1 \cdot 5 - (i \sqrt{167})}{6}$

Step 2.7.6

Factor $- 1$ out of $- 1 (5) - (i \sqrt{167})$ .

$x = \frac{- 1 (5 + i \sqrt{167})}{6}$

Step 2.7.7

Move the negative in front of the fraction.

$x = - \frac{5 + i \sqrt{167}}{6}$

Step 2.8

The final answer is the combination of both solutions.

$x = - \frac{5 - i \sqrt{167}}{6}, - \frac{5 + i \sqrt{167}}{6}$