Find All Complex Solutions 2x^2+3x-11=0

$2 x^{2} + 3 x - 11 = 0$

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 $a = 2$ , $b = 3$ , and $c = - 11$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (2 \cdot - 11)}}{2 \cdot 2}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 11}}{2 \cdot 2}$

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

Multiply $- 4$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 8 \cdot - 11}}{2 \cdot 2}$

Multiply $- 8$ by $- 11$ .

$x = \frac{- 3 \pm \sqrt{9 + 88}}{2 \cdot 2}$

Add $9$ and $88$ .

$x = \frac{- 3 \pm \sqrt{97}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 3 \pm \sqrt{97}}{4}$

Step 4

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{97}}{4}, - \frac{3 + \sqrt{97}}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 - \sqrt{97}}{4}, - \frac{3 + \sqrt{97}}{4}$

Decimal Form:

$x = 1.71221445 \dots, - 3.21221445 \dots$