Find All Complex Solutions 8x^2-5x+3=0

$8 x^{2} - 5 x + 3 = 0$

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 $a = 8$ , $b = - 5$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (8 \cdot 3)}}{2 \cdot 8}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 8 \cdot 3}}{2 \cdot 8}$

Multiply $- 4 \cdot 8 \cdot 3$ .

Multiply $- 4$ by $8$ .

$x = \frac{5 \pm \sqrt{25 - 32 \cdot 3}}{2 \cdot 8}$

Multiply $- 32$ by $3$ .

$x = \frac{5 \pm \sqrt{25 - 96}}{2 \cdot 8}$

Subtract $96$ from $25$ .

$x = \frac{5 \pm \sqrt{- 71}}{2 \cdot 8}$

Rewrite $- 71$ as $- 1 (71)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 71}}{2 \cdot 8}$

Rewrite $\sqrt{- 1 (71)}$ as $\sqrt{- 1} \cdot \sqrt{71}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{71}}{2 \cdot 8}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{5 \pm i \sqrt{71}}{2 \cdot 8}$

Multiply $2$ by $8$ .

$x = \frac{5 \pm i \sqrt{71}}{16}$