Find All Complex Solutions 74x^2+1=10x

$74 x^{2} + 1 = 10 x$

Step 1

Subtract $10 x$ from both sides of the equation.

$74 x^{2} + 1 - 10 x = 0$

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 $a = 74$ , $b = - 10$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (74 \cdot 1)}}{2 \cdot 74}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 74 \cdot 1}}{2 \cdot 74}$

Multiply $- 4 \cdot 74 \cdot 1$ .

Multiply $- 4$ by $74$ .

$x = \frac{10 \pm \sqrt{100 - 296 \cdot 1}}{2 \cdot 74}$

Multiply $- 296$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 296}}{2 \cdot 74}$

Subtract $296$ from $100$ .

$x = \frac{10 \pm \sqrt{- 196}}{2 \cdot 74}$

Rewrite $- 196$ as $- 1 (196)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 196}}{2 \cdot 74}$

Rewrite $\sqrt{- 1 (196)}$ as $\sqrt{- 1} \cdot \sqrt{196}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{196}}{2 \cdot 74}$

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

$x = \frac{10 \pm i \cdot \sqrt{196}}{2 \cdot 74}$

Rewrite $196$ as $14^{2}$ .

$x = \frac{10 \pm i \cdot \sqrt{14^{2}}}{2 \cdot 74}$

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

$x = \frac{10 \pm i \cdot 14}{2 \cdot 74}$

Move $14$ to the left of $i$ .

$x = \frac{10 \pm 14 i}{2 \cdot 74}$

Multiply $2$ by $74$ .

$x = \frac{10 \pm 14 i}{148}$

Simplify $\frac{10 \pm 14 i}{148}$ .

$x = \frac{5 \pm 7 i}{74}$