Solve Using the Quadratic Formula a^2-8a-11=0

$a^{2} - 8 a - 11 = 0$

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 $a = 1$ , $b = - 8$ , and $c = - 11$ into the quadratic formula and solve for $a$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot - 11)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$a = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 11}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 11$ .

Multiply $- 4$ by $1$ .

$a = \frac{8 \pm \sqrt{64 - 4 \cdot - 11}}{2 \cdot 1}$

Multiply $- 4$ by $- 11$ .

$a = \frac{8 \pm \sqrt{64 + 44}}{2 \cdot 1}$

Add $64$ and $44$ .

$a = \frac{8 \pm \sqrt{108}}{2 \cdot 1}$

Rewrite $108$ as $6^{2} \cdot 3$ .

Factor $36$ out of $108$ .

$a = \frac{8 \pm \sqrt{36 (3)}}{2 \cdot 1}$

Rewrite $36$ as $6^{2}$ .

$a = \frac{8 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$a = \frac{8 \pm 6 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$a = \frac{8 \pm 6 \sqrt{3}}{2}$

Simplify $\frac{8 \pm 6 \sqrt{3}}{2}$ .

$a = 4 \pm 3 \sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$a = 4 \pm 3 \sqrt{3}$

Decimal Form:

$a = 9.19615242 \dots, - 1.19615242 \dots$