Find All Complex Solutions 4x^2+8x-77=0

$4 x^{2} + 8 x - 77 = 0$

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 $a = 4$ , $b = 8$ , and $c = - 77$ into the quadratic formula and solve for $x$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (4 \cdot - 77)}}{2 \cdot 4}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 4 \cdot - 77}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot - 77$ .

Multiply $- 4$ by $4$ .

$x = \frac{- 8 \pm \sqrt{64 - 16 \cdot - 77}}{2 \cdot 4}$

Multiply $- 16$ by $- 77$ .

$x = \frac{- 8 \pm \sqrt{64 + 1232}}{2 \cdot 4}$

Add $64$ and $1232$ .

$x = \frac{- 8 \pm \sqrt{1296}}{2 \cdot 4}$

Rewrite $1296$ as $36^{2}$ .

$x = \frac{- 8 \pm \sqrt{36^{2}}}{2 \cdot 4}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 8 \pm 36}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{- 8 \pm 36}{8}$

Simplify $\frac{- 8 \pm 36}{8}$ .

$x = \frac{- 2 \pm 9}{2}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{7}{2}, - \frac{11}{2}$