Find All Complex Solutions 5x^2-6x+5=0

$5 x^{2} - 6 x + 5 = 0$

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 $a = 5$ , $b = - 6$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (5 \cdot 5)}}{2 \cdot 5}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 5 \cdot 5}}{2 \cdot 5}$

Multiply $- 4 \cdot 5 \cdot 5$ .

Multiply $- 4$ by $5$ .

$x = \frac{6 \pm \sqrt{36 - 20 \cdot 5}}{2 \cdot 5}$

Multiply $- 20$ by $5$ .

$x = \frac{6 \pm \sqrt{36 - 100}}{2 \cdot 5}$

Subtract $100$ from $36$ .

$x = \frac{6 \pm \sqrt{- 64}}{2 \cdot 5}$

Rewrite $- 64$ as $- 1 (64)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 64}}{2 \cdot 5}$

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{64}}{2 \cdot 5}$

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

$x = \frac{6 \pm i \cdot \sqrt{64}}{2 \cdot 5}$

Rewrite $64$ as $8^{2}$ .

$x = \frac{6 \pm i \cdot \sqrt{8^{2}}}{2 \cdot 5}$

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

$x = \frac{6 \pm i \cdot 8}{2 \cdot 5}$

Move $8$ to the left of $i$ .

$x = \frac{6 \pm 8 i}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{6 \pm 8 i}{10}$

Simplify $\frac{6 \pm 8 i}{10}$ .

$x = \frac{3 \pm 4 i}{5}$