Pre-Algebra Examples

$x^{2} + 13 + 36 = (x + 9)$

Step 1

Add $13$ and $36$ .

$x^{2} + 49 = x + 9$

Step 2

Subtract $x$ from both sides of the equation.

$x^{2} + 49 - x = 9$

Step 3

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

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Step 3.1

Subtract $9$ from both sides of the equation.

$x^{2} + 49 - x - 9 = 0$

Step 3.2

Subtract $9$ from $49$ .

$x^{2} - x + 40 = 0$

Step 4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5

Substitute the values $a = 1$ , $b = - 1$ , and $c = 40$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot 40)}}{2 \cdot 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 40}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot 40$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 40}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $40$ .

$x = \frac{1 \pm \sqrt{1 - 160}}{2 \cdot 1}$

Step 6.1.3

Subtract $160$ from $1$ .

$x = \frac{1 \pm \sqrt{- 159}}{2 \cdot 1}$

Step 6.1.4

Rewrite $- 159$ as $- 1 (159)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 159}}{2 \cdot 1}$

Step 6.1.5

Rewrite $\sqrt{- 1 (159)}$ as $\sqrt{- 1} \cdot \sqrt{159}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{159}}{2 \cdot 1}$

Step 6.1.6

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

$x = \frac{1 \pm i \sqrt{159}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{159}}{2}$

Step 7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 40}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 40$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 40}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $40$ .

$x = \frac{1 \pm \sqrt{1 - 160}}{2 \cdot 1}$

Step 7.1.3

Subtract $160$ from $1$ .

$x = \frac{1 \pm \sqrt{- 159}}{2 \cdot 1}$

Step 7.1.4

Rewrite $- 159$ as $- 1 (159)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 159}}{2 \cdot 1}$

Step 7.1.5

Rewrite $\sqrt{- 1 (159)}$ as $\sqrt{- 1} \cdot \sqrt{159}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{159}}{2 \cdot 1}$

Step 7.1.6

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

$x = \frac{1 \pm i \sqrt{159}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{159}}{2}$

Step 7.3

Change the $\pm$ to $+$ .

$x = \frac{1 + i \sqrt{159}}{2}$

Step 8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 40}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot 40$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 40}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $40$ .

$x = \frac{1 \pm \sqrt{1 - 160}}{2 \cdot 1}$

Step 8.1.3

Subtract $160$ from $1$ .

$x = \frac{1 \pm \sqrt{- 159}}{2 \cdot 1}$

Step 8.1.4

Rewrite $- 159$ as $- 1 (159)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 159}}{2 \cdot 1}$

Step 8.1.5

Rewrite $\sqrt{- 1 (159)}$ as $\sqrt{- 1} \cdot \sqrt{159}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{159}}{2 \cdot 1}$

Step 8.1.6

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

$x = \frac{1 \pm i \sqrt{159}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{159}}{2}$

Step 8.3

Change the $\pm$ to $-$ .

$x = \frac{1 - i \sqrt{159}}{2}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{159}}{2}, \frac{1 - i \sqrt{159}}{2}$