Trigonometry Examples

$6 x^{2} - 4 x - 12 = 0$

Step 1

Factor $2$ out of $6 x^{2} - 4 x - 12$ .

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Step 1.1

Factor $2$ out of $6 x^{2}$ .

$2 (3 x^{2}) - 4 x - 12 = 0$

Step 1.2

Factor $2$ out of $- 4 x$ .

$2 (3 x^{2}) + 2 (- 2 x) - 12 = 0$

Step 1.3

Factor $2$ out of $- 12$ .

$2 (3 x^{2}) + 2 (- 2 x) + 2 (- 6) = 0$

Step 1.4

Factor $2$ out of $2 (3 x^{2}) + 2 (- 2 x)$ .

$2 (3 x^{2} - 2 x) + 2 (- 6) = 0$

Step 1.5

Factor $2$ out of $2 (3 x^{2} - 2 x) + 2 (- 6)$ .

$2 (3 x^{2} - 2 x - 6) = 0$

Step 2

Divide each term in $2 (3 x^{2} - 2 x - 6) = 0$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 (3 x^{2} - 2 x - 6) = 0$ by $2$ .

$\frac{2 (3 x^{2} - 2 x - 6)}{2} = \frac{0}{2}$

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.

$\frac{2 (3 x^{2} - 2 x - 6)}{2} = \frac{0}{2}$

Step 2.2.1.2

Divide $3 x^{2} - 2 x - 6$ by $1$ .

$3 x^{2} - 2 x - 6 = \frac{0}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $2$ .

$3 x^{2} - 2 x - 6 = 0$

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 $a = 3$ , $b = - 2$ , and $c = - 6$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (3 \cdot - 6)}}{2 \cdot 3}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 3 \cdot - 6}}{2 \cdot 3}$

Step 5.1.2

Multiply $- 4 \cdot 3 \cdot - 6$ .

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Step 5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot - 6}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $- 6$ .

$x = \frac{2 \pm \sqrt{4 + 72}}{2 \cdot 3}$

Step 5.1.3

Add $4$ and $72$ .

$x = \frac{2 \pm \sqrt{76}}{2 \cdot 3}$

Step 5.1.4

Rewrite $76$ as $2^{2} \cdot 19$ .

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Step 5.1.4.1

Factor $4$ out of $76$ .

$x = \frac{2 \pm \sqrt{4 (19)}}{2 \cdot 3}$

Step 5.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 3}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{19}}{2 \cdot 3}$

Step 5.2

Multiply $2$ by $3$ .

$x = \frac{2 \pm 2 \sqrt{19}}{6}$

Step 5.3

Simplify $\frac{2 \pm 2 \sqrt{19}}{6}$ .

$x = \frac{1 \pm \sqrt{19}}{3}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{19}}{3}, \frac{1 - \sqrt{19}}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{19}}{3}, \frac{1 - \sqrt{19}}{3}$

Decimal Form:

$x = 1.78629964 \dots, - 1.11963298 \dots$