Solve Using the Quadratic Formula 9x^2-6x-11=0

$9 x^{2} - 6 x - 11 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 9$ , $b = - 6$ , and $c = - 11$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (9 \cdot - 11)}}{2 \cdot 9}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 9 \cdot - 11}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot - 11$ .

Multiply $- 4$ by $9$ .

$x = \frac{6 \pm \sqrt{36 - 36 \cdot - 11}}{2 \cdot 9}$

Multiply $- 36$ by $- 11$ .

$x = \frac{6 \pm \sqrt{36 + 396}}{2 \cdot 9}$

Add $36$ and $396$ .

$x = \frac{6 \pm \sqrt{432}}{2 \cdot 9}$

Rewrite $432$ as $12^{2} \cdot 3$ .

Factor $144$ out of $432$ .

$x = \frac{6 \pm \sqrt{144 (3)}}{2 \cdot 9}$

Rewrite $144$ as $12^{2}$ .

$x = \frac{6 \pm \sqrt{12^{2} \cdot 3}}{2 \cdot 9}$

Pull terms out from under the radical.

$x = \frac{6 \pm 12 \sqrt{3}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{6 \pm 12 \sqrt{3}}{18}$

Simplify $\frac{6 \pm 12 \sqrt{3}}{18}$ .

$x = \frac{1 \pm 2 \sqrt{3}}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 \pm 2 \sqrt{3}}{3}$

Decimal Form:

$x = 1.48803387 \dots, - 0.82136720 \dots$