Solve Using the Quadratic Formula -4x^2=-12x+11

$- 4 x^{2} = - 12 x + 11$

Step 1

Move all the expressions to the left side of the equation.

Add $12 x$ to both sides of the equation.

$- 4 x^{2} + 12 x = 11$

Subtract $11$ from both sides of the equation.

$- 4 x^{2} + 12 x - 11 = 0$

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 $a = - 4$ , $b = 12$ , and $c = - 11$ into the quadratic formula and solve for $x$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (- 4 \cdot - 11)}}{2 \cdot - 4}$

Step 4

Simplify.

Simplify the numerator.

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 4 \cdot - 11}}{2 \cdot - 4}$

Multiply $- 4 \cdot - 4 \cdot - 11$ .

Multiply $- 4$ by $- 4$ .

$x = \frac{- 12 \pm \sqrt{144 + 16 \cdot - 11}}{2 \cdot - 4}$

Multiply $16$ by $- 11$ .

$x = \frac{- 12 \pm \sqrt{144 - 176}}{2 \cdot - 4}$

Subtract $176$ from $144$ .

$x = \frac{- 12 \pm \sqrt{- 32}}{2 \cdot - 4}$

Rewrite $- 32$ as $- 1 (32)$ .

$x = \frac{- 12 \pm \sqrt{- 1 \cdot 32}}{2 \cdot - 4}$

Rewrite $\sqrt{- 1 (32)}$ as $\sqrt{- 1} \cdot \sqrt{32}$ .

$x = \frac{- 12 \pm \sqrt{- 1} \cdot \sqrt{32}}{2 \cdot - 4}$

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

$x = \frac{- 12 \pm i \cdot \sqrt{32}}{2 \cdot - 4}$

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$x = \frac{- 12 \pm i \cdot \sqrt{16 (2)}}{2 \cdot - 4}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 12 \pm i \cdot \sqrt{4^{2} \cdot 2}}{2 \cdot - 4}$

Pull terms out from under the radical.

$x = \frac{- 12 \pm i \cdot (4 \sqrt{2})}{2 \cdot - 4}$

Move $4$ to the left of $i$ .

$x = \frac{- 12 \pm 4 i \sqrt{2}}{2 \cdot - 4}$

Multiply $2$ by $- 4$ .

$x = \frac{- 12 \pm 4 i \sqrt{2}}{- 8}$

Simplify $\frac{- 12 \pm 4 i \sqrt{2}}{- 8}$ .

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