Trigonometry Examples

$3 m^{2} + 2 m = 9$

Step 1

Subtract $9$ from both sides of the equation.

$3 m^{2} + 2 m - 9 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 3$ , $b = 2$ , and $c = - 9$ into the quadratic formula and solve for $m$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (3 \cdot - 9)}}{2 \cdot 3}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$m = \frac{- 2 \pm \sqrt{4 - 4 \cdot 3 \cdot - 9}}{2 \cdot 3}$

Step 4.1.2

Multiply $- 4 \cdot 3 \cdot - 9$ .

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Step 4.1.2.1

Multiply $- 4$ by $3$ .

$m = \frac{- 2 \pm \sqrt{4 - 12 \cdot - 9}}{2 \cdot 3}$

Step 4.1.2.2

Multiply $- 12$ by $- 9$ .

$m = \frac{- 2 \pm \sqrt{4 + 108}}{2 \cdot 3}$

Step 4.1.3

Add $4$ and $108$ .

$m = \frac{- 2 \pm \sqrt{112}}{2 \cdot 3}$

Step 4.1.4

Rewrite $112$ as $4^{2} \cdot 7$ .

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Step 4.1.4.1

Factor $16$ out of $112$ .

$m = \frac{- 2 \pm \sqrt{16 (7)}}{2 \cdot 3}$

Step 4.1.4.2

Rewrite $16$ as $4^{2}$ .

$m = \frac{- 2 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 3}$

Step 4.1.5

Pull terms out from under the radical.

$m = \frac{- 2 \pm 4 \sqrt{7}}{2 \cdot 3}$

Step 4.2

Multiply $2$ by $3$ .

$m = \frac{- 2 \pm 4 \sqrt{7}}{6}$

Step 4.3

Simplify $\frac{- 2 \pm 4 \sqrt{7}}{6}$ .

$m = \frac{- 1 \pm 2 \sqrt{7}}{3}$

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 $2$ to the power of $2$ .

$m = \frac{- 2 \pm \sqrt{4 - 4 \cdot 3 \cdot - 9}}{2 \cdot 3}$

Step 5.1.2

Multiply $- 4 \cdot 3 \cdot - 9$ .

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Step 5.1.2.1

Multiply $- 4$ by $3$ .

$m = \frac{- 2 \pm \sqrt{4 - 12 \cdot - 9}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $- 9$ .

$m = \frac{- 2 \pm \sqrt{4 + 108}}{2 \cdot 3}$

Step 5.1.3

Add $4$ and $108$ .

$m = \frac{- 2 \pm \sqrt{112}}{2 \cdot 3}$

Step 5.1.4

Rewrite $112$ as $4^{2} \cdot 7$ .

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Step 5.1.4.1

Factor $16$ out of $112$ .

$m = \frac{- 2 \pm \sqrt{16 (7)}}{2 \cdot 3}$

Step 5.1.4.2

Rewrite $16$ as $4^{2}$ .

$m = \frac{- 2 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 3}$

Step 5.1.5

Pull terms out from under the radical.

$m = \frac{- 2 \pm 4 \sqrt{7}}{2 \cdot 3}$

Step 5.2

Multiply $2$ by $3$ .

$m = \frac{- 2 \pm 4 \sqrt{7}}{6}$

Step 5.3

Simplify $\frac{- 2 \pm 4 \sqrt{7}}{6}$ .

$m = \frac{- 1 \pm 2 \sqrt{7}}{3}$

Step 5.4

Change the $\pm$ to $+$ .

$m = \frac{- 1 + 2 \sqrt{7}}{3}$

Step 5.5

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

$m = \frac{- 1 \cdot 1 + 2 \sqrt{7}}{3}$

Step 5.6

Factor $- 1$ out of $2 \sqrt{7}$ .

$m = \frac{- 1 \cdot 1 - (- 2 \sqrt{7})}{3}$

Step 5.7

Factor $- 1$ out of $- 1 (1) - (- 2 \sqrt{7})$ .

$m = \frac{- 1 (1 - 2 \sqrt{7})}{3}$

Step 5.8

Move the negative in front of the fraction.

$m = - \frac{1 - 2 \sqrt{7}}{3}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$m = \frac{- 2 \pm \sqrt{4 - 4 \cdot 3 \cdot - 9}}{2 \cdot 3}$

Step 6.1.2

Multiply $- 4 \cdot 3 \cdot - 9$ .

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Step 6.1.2.1

Multiply $- 4$ by $3$ .

$m = \frac{- 2 \pm \sqrt{4 - 12 \cdot - 9}}{2 \cdot 3}$

Step 6.1.2.2

Multiply $- 12$ by $- 9$ .

$m = \frac{- 2 \pm \sqrt{4 + 108}}{2 \cdot 3}$

Step 6.1.3

Add $4$ and $108$ .

$m = \frac{- 2 \pm \sqrt{112}}{2 \cdot 3}$

Step 6.1.4

Rewrite $112$ as $4^{2} \cdot 7$ .

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Step 6.1.4.1

Factor $16$ out of $112$ .

$m = \frac{- 2 \pm \sqrt{16 (7)}}{2 \cdot 3}$

Step 6.1.4.2

Rewrite $16$ as $4^{2}$ .

$m = \frac{- 2 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 3}$

Step 6.1.5

Pull terms out from under the radical.

$m = \frac{- 2 \pm 4 \sqrt{7}}{2 \cdot 3}$

Step 6.2

Multiply $2$ by $3$ .

$m = \frac{- 2 \pm 4 \sqrt{7}}{6}$

Step 6.3

Simplify $\frac{- 2 \pm 4 \sqrt{7}}{6}$ .

$m = \frac{- 1 \pm 2 \sqrt{7}}{3}$

Step 6.4

Change the $\pm$ to $-$ .

$m = \frac{- 1 - 2 \sqrt{7}}{3}$

Step 6.5

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

$m = \frac{- 1 \cdot 1 - 2 \sqrt{7}}{3}$

Step 6.6

Factor $- 1$ out of $- 2 \sqrt{7}$ .

$m = \frac{- 1 \cdot 1 - (2 \sqrt{7})}{3}$

Step 6.7

Factor $- 1$ out of $- 1 (1) - (2 \sqrt{7})$ .

$m = \frac{- 1 (1 + 2 \sqrt{7})}{3}$

Step 6.8

Move the negative in front of the fraction.

$m = - \frac{1 + 2 \sqrt{7}}{3}$

Step 7

The final answer is the combination of both solutions.

$m = - \frac{1 - 2 \sqrt{7}}{3}, - \frac{1 + 2 \sqrt{7}}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$m = - \frac{1 - 2 \sqrt{7}}{3}, - \frac{1 + 2 \sqrt{7}}{3}$

Decimal Form:

$m = 1.43050087 \dots, - 2.09716754 \dots$