Find All Complex Solutions 89x^2+1=16x

$89 x^{2} + 1 = 16 x$

Step 1

Subtract $16 x$ from both sides of the equation.

$89 x^{2} + 1 - 16 x = 0$

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 $a = 89$ , $b = - 16$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (89 \cdot 1)}}{2 \cdot 89}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 16$ to the power of $2$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 89 \cdot 1}}{2 \cdot 89}$

Multiply $- 4 \cdot 89 \cdot 1$ .

Multiply $- 4$ by $89$ .

$x = \frac{16 \pm \sqrt{256 - 356 \cdot 1}}{2 \cdot 89}$

Multiply $- 356$ by $1$ .

$x = \frac{16 \pm \sqrt{256 - 356}}{2 \cdot 89}$

Subtract $356$ from $256$ .

$x = \frac{16 \pm \sqrt{- 100}}{2 \cdot 89}$

Rewrite $- 100$ as $- 1 (100)$ .

$x = \frac{16 \pm \sqrt{- 1 \cdot 100}}{2 \cdot 89}$

Rewrite $\sqrt{- 1 (100)}$ as $\sqrt{- 1} \cdot \sqrt{100}$ .

$x = \frac{16 \pm \sqrt{- 1} \cdot \sqrt{100}}{2 \cdot 89}$

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

$x = \frac{16 \pm i \cdot \sqrt{100}}{2 \cdot 89}$

Rewrite $100$ as $10^{2}$ .

$x = \frac{16 \pm i \cdot \sqrt{10^{2}}}{2 \cdot 89}$

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

$x = \frac{16 \pm i \cdot 10}{2 \cdot 89}$

Move $10$ to the left of $i$ .

$x = \frac{16 \pm 10 i}{2 \cdot 89}$

Multiply $2$ by $89$ .

$x = \frac{16 \pm 10 i}{178}$

Simplify $\frac{16 \pm 10 i}{178}$ .

$x = \frac{8 \pm 5 i}{89}$