Solve Using the Quadratic Formula 2x=4x+22+x^2+7x

$2 x = 4 x + 22 + x^{2} + 7 x$

Step 1

Move all terms to the left side of the equation and simplify.

Simplify the right side.

Add $4 x$ and $7 x$ .

$2 x = 11 x + 22 + x^{2}$

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

Subtract $11 x$ from both sides of the equation.

$2 x - 11 x = 22 + x^{2}$

Subtract $22$ from both sides of the equation.

$2 x - 11 x - 22 = x^{2}$

Subtract $x^{2}$ from both sides of the equation.

$2 x - 11 x - 22 - x^{2} = 0$

Subtract $11 x$ from $2 x$ .

$- 9 x - 22 - x^{2} = 0$

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 $a = - 1$ , $b = - 9$ , and $c = - 22$ into the quadratic formula and solve for $x$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (- 1 \cdot - 22)}}{2 \cdot - 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot - 1 \cdot - 22}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 22$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{9 \pm \sqrt{81 + 4 \cdot - 22}}{2 \cdot - 1}$

Multiply $4$ by $- 22$ .

$x = \frac{9 \pm \sqrt{81 - 88}}{2 \cdot - 1}$

Subtract $88$ from $81$ .

$x = \frac{9 \pm \sqrt{- 7}}{2 \cdot - 1}$

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{9 \pm \sqrt{- 1 \cdot 7}}{2 \cdot - 1}$

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{9 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot - 1}$

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

$x = \frac{9 \pm i \sqrt{7}}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$x = \frac{9 \pm i \sqrt{7}}{- 2}$

Move the negative in front of the fraction.

$x = - \frac{9 \pm i \sqrt{7}}{2}$