Trigonometry Examples

$f (x) = 0$ , $f (x) = - 6 x^{2} + 9 x + 9$

Step 1

Substitute $- 6 x^{2} + 9 x + 9$ for $f (x)$ .

$- 6 x^{2} + 9 x + 9 = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Factor $- 3$ out of $- 6 x^{2} + 9 x + 9$ .

Tap for more steps...

Step 2.1.1

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

$- 3 (2 x^{2}) + 9 x + 9 = 0$

Step 2.1.2

Factor $- 3$ out of $9 x$ .

$- 3 (2 x^{2}) - 3 (- 3 x) + 9 = 0$

Step 2.1.3

Factor $- 3$ out of $9$ .

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

Step 2.1.4

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

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

Step 2.1.5

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

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Cancel the common factor of $- 3$ .

Tap for more steps...

Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

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

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

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Divide $0$ by $- 3$ .

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

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

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

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

Simplify the numerator.

Tap for more steps...

Step 2.5.1.1

Raise $- 3$ to the power of $2$ .

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

Step 2.5.1.2

Multiply $- 4 \cdot 2 \cdot - 3$ .

Tap for more steps...

Step 2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{3 \pm \sqrt{9 - 8 \cdot - 3}}{2 \cdot 2}$

Step 2.5.1.2.2

Multiply $- 8$ by $- 3$ .

$x = \frac{3 \pm \sqrt{9 + 24}}{2 \cdot 2}$

Step 2.5.1.3

Add $9$ and $24$ .

$x = \frac{3 \pm \sqrt{33}}{2 \cdot 2}$

Step 2.5.2

Multiply $2$ by $2$ .

$x = \frac{3 \pm \sqrt{33}}{4}$

Step 2.6

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

Tap for more steps...

Step 2.6.1

Simplify the numerator.

Tap for more steps...

Step 2.6.1.1

Raise $- 3$ to the power of $2$ .

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

Step 2.6.1.2

Multiply $- 4 \cdot 2 \cdot - 3$ .

Tap for more steps...

Step 2.6.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{3 \pm \sqrt{9 - 8 \cdot - 3}}{2 \cdot 2}$

Step 2.6.1.2.2

Multiply $- 8$ by $- 3$ .

$x = \frac{3 \pm \sqrt{9 + 24}}{2 \cdot 2}$

Step 2.6.1.3

Add $9$ and $24$ .

$x = \frac{3 \pm \sqrt{33}}{2 \cdot 2}$

Step 2.6.2

Multiply $2$ by $2$ .

$x = \frac{3 \pm \sqrt{33}}{4}$

Step 2.6.3

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{33}}{4}$

Step 2.7

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

Tap for more steps...

Step 2.7.1

Simplify the numerator.

Tap for more steps...

Step 2.7.1.1

Raise $- 3$ to the power of $2$ .

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

Step 2.7.1.2

Multiply $- 4 \cdot 2 \cdot - 3$ .

Tap for more steps...

Step 2.7.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{3 \pm \sqrt{9 - 8 \cdot - 3}}{2 \cdot 2}$

Step 2.7.1.2.2

Multiply $- 8$ by $- 3$ .

$x = \frac{3 \pm \sqrt{9 + 24}}{2 \cdot 2}$

Step 2.7.1.3

Add $9$ and $24$ .

$x = \frac{3 \pm \sqrt{33}}{2 \cdot 2}$

Step 2.7.2

Multiply $2$ by $2$ .

$x = \frac{3 \pm \sqrt{33}}{4}$

Step 2.7.3

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{33}}{4}$

Step 2.8

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{33}}{4}, \frac{3 - \sqrt{33}}{4}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{33}}{4}, \frac{3 - \sqrt{33}}{4}$

Decimal Form:

$x = 2.18614066 \dots, - 0.68614066 \dots$