Solve Using the Quadratic Formula 81x^2-18x+2=0

$81 x^{2} - 18 x + 2 = 0$

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 $a = 81$ , $b = - 18$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{18 \pm \sqrt{{(- 18)}^{2} - 4 \cdot (81 \cdot 2)}}{2 \cdot 81}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 18$ to the power of $2$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 81 \cdot 2}}{2 \cdot 81}$

Multiply $- 4 \cdot 81 \cdot 2$ .

Multiply $- 4$ by $81$ .

$x = \frac{18 \pm \sqrt{324 - 324 \cdot 2}}{2 \cdot 81}$

Multiply $- 324$ by $2$ .

$x = \frac{18 \pm \sqrt{324 - 648}}{2 \cdot 81}$

Subtract $648$ from $324$ .

$x = \frac{18 \pm \sqrt{- 324}}{2 \cdot 81}$

Rewrite $- 324$ as $- 1 (324)$ .

$x = \frac{18 \pm \sqrt{- 1 \cdot 324}}{2 \cdot 81}$

Rewrite $\sqrt{- 1 (324)}$ as $\sqrt{- 1} \cdot \sqrt{324}$ .

$x = \frac{18 \pm \sqrt{- 1} \cdot \sqrt{324}}{2 \cdot 81}$

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

$x = \frac{18 \pm i \cdot \sqrt{324}}{2 \cdot 81}$

Rewrite $324$ as $18^{2}$ .

$x = \frac{18 \pm i \cdot \sqrt{18^{2}}}{2 \cdot 81}$

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

$x = \frac{18 \pm i \cdot 18}{2 \cdot 81}$

Move $18$ to the left of $i$ .

$x = \frac{18 \pm 18 i}{2 \cdot 81}$

Multiply $2$ by $81$ .

$x = \frac{18 \pm 18 i}{162}$

Simplify $\frac{18 \pm 18 i}{162}$ .

$x = \frac{1 \pm i}{9}$