Solve Using the Quadratic Formula 2x^2+10x+11=0

$2 x^{2} + 10 x + 11 = 0$

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 $a = 2$ , $b = 10$ , and $c = 11$ into the quadratic formula and solve for $x$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (2 \cdot 11)}}{2 \cdot 2}$

Step 3

Simplify.

Simplify the numerator.

Raise $10$ to the power of $2$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 2 \cdot 11}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 11$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 10 \pm \sqrt{100 - 8 \cdot 11}}{2 \cdot 2}$

Multiply $- 8$ by $11$ .

$x = \frac{- 10 \pm \sqrt{100 - 88}}{2 \cdot 2}$

Subtract $88$ from $100$ .

$x = \frac{- 10 \pm \sqrt{12}}{2 \cdot 2}$

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

Factor $4$ out of $12$ .

$x = \frac{- 10 \pm \sqrt{4 (3)}}{2 \cdot 2}$

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

$x = \frac{- 10 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 2}$

Pull terms out from under the radical.

$x = \frac{- 10 \pm 2 \sqrt{3}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 10 \pm 2 \sqrt{3}}{4}$

Simplify $\frac{- 10 \pm 2 \sqrt{3}}{4}$ .

$x = \frac{- 5 \pm \sqrt{3}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{- 5 \pm \sqrt{3}}{2}$

Decimal Form:

$x = - 1.63397459 \dots, - 3.36602540 \dots$