Find All Complex Solutions 52x^2+1=8x

$52 x^{2} + 1 = 8 x$

Step 1

Subtract $8 x$ from both sides of the equation.

$52 x^{2} + 1 - 8 x = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 52$ , $b = - 8$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (52 \cdot 1)}}{2 \cdot 52}$

Step 4

Simplify.

Simplify the numerator.

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 52 \cdot 1}}{2 \cdot 52}$

Multiply $- 4 \cdot 52 \cdot 1$ .

Multiply $- 4$ by $52$ .

$x = \frac{8 \pm \sqrt{64 - 208 \cdot 1}}{2 \cdot 52}$

Multiply $- 208$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 208}}{2 \cdot 52}$

Subtract $208$ from $64$ .

$x = \frac{8 \pm \sqrt{- 144}}{2 \cdot 52}$

Rewrite $- 144$ as $- 1 (144)$ .

$x = \frac{8 \pm \sqrt{- 1 \cdot 144}}{2 \cdot 52}$

Rewrite $\sqrt{- 1 (144)}$ as $\sqrt{- 1} \cdot \sqrt{144}$ .

$x = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{144}}{2 \cdot 52}$

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

$x = \frac{8 \pm i \cdot \sqrt{144}}{2 \cdot 52}$

Rewrite $144$ as $12^{2}$ .

$x = \frac{8 \pm i \cdot \sqrt{12^{2}}}{2 \cdot 52}$

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

$x = \frac{8 \pm i \cdot 12}{2 \cdot 52}$

Move $12$ to the left of $i$ .

$x = \frac{8 \pm 12 i}{2 \cdot 52}$

Multiply $2$ by $52$ .

$x = \frac{8 \pm 12 i}{104}$

Simplify $\frac{8 \pm 12 i}{104}$ .

$x = \frac{2 \pm 3 i}{26}$