Algebra Examples

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

Step 1

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

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Step 1.1

Simplify the right side.

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Step 1.1.1

Add $4 x$ and $7 x$ .

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Subtract $22$ from both sides of the equation.

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

Step 1.2.3

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

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

Step 1.3

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.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

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

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Step 4.1.2.1

Multiply $- 4$ by $- 1$ .

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

Step 4.1.2.2

Multiply $4$ by $- 22$ .

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

Step 4.1.3

Subtract $88$ from $81$ .

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

Step 4.1.4

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

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

Step 4.1.5

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

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

Step 4.1.6

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

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

Step 4.2

Multiply $2$ by $- 1$ .

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

Step 4.3

Move the negative in front of the fraction.

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