Find All Complex Solutions 16x^2+62x+42=0

$16 x^{2} + 62 x + 42 = 0$

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 $a = 16$ , $b = 62$ , and $c = 42$ into the quadratic formula and solve for $x$ .

$\frac{- 62 \pm \sqrt{62^{2} - 4 \cdot (16 \cdot 42)}}{2 \cdot 16}$

Step 3

Simplify.

Simplify the numerator.

Raise $62$ to the power of $2$ .

$x = \frac{- 62 \pm \sqrt{3844 - 4 \cdot 16 \cdot 42}}{2 \cdot 16}$

Multiply $- 4 \cdot 16 \cdot 42$ .

Multiply $- 4$ by $16$ .

$x = \frac{- 62 \pm \sqrt{3844 - 64 \cdot 42}}{2 \cdot 16}$

Multiply $- 64$ by $42$ .

$x = \frac{- 62 \pm \sqrt{3844 - 2688}}{2 \cdot 16}$

Subtract $2688$ from $3844$ .

$x = \frac{- 62 \pm \sqrt{1156}}{2 \cdot 16}$

Rewrite $1156$ as $34^{2}$ .

$x = \frac{- 62 \pm \sqrt{34^{2}}}{2 \cdot 16}$

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

$x = \frac{- 62 \pm 34}{2 \cdot 16}$

Multiply $2$ by $16$ .

$x = \frac{- 62 \pm 34}{32}$

Simplify $\frac{- 62 \pm 34}{32}$ .

$x = \frac{- 31 \pm 17}{16}$

Step 4

The final answer is the combination of both solutions.

$x = - \frac{7}{8}, - 3$