Algebra Examples

$x^{2} + 17 x + 9 = 13 x - 3$

Step 1

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

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Step 1.1

Subtract $13 x$ from both sides of the equation.

$x^{2} + 17 x + 9 - 13 x = - 3$

Step 1.2

Add $3$ to both sides of the equation.

$x^{2} + 17 x + 9 - 13 x + 3 = 0$

Step 2

Simplify $x^{2} + 17 x + 9 - 13 x + 3$ .

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Step 2.1

Subtract $13 x$ from $17 x$ .

$x^{2} + 4 x + 9 + 3 = 0$

Step 2.2

Add $9$ and $3$ .

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

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 1$ , $b = 4$ , and $c = 12$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot 12)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 12}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 12$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 12}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $12$ .

$x = \frac{- 4 \pm \sqrt{16 - 48}}{2 \cdot 1}$

Step 5.1.3

Subtract $48$ from $16$ .

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

Step 5.1.4

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

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

Step 5.1.5

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

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

Step 5.1.6

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

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

Step 5.1.7

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

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Step 5.1.7.1

Factor $16$ out of $32$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (2)}}{2 \cdot 1}$

Step 5.1.7.2

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

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

Step 5.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (4 \sqrt{2})}{2 \cdot 1}$

Step 5.1.9

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

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

Step 5.2

Multiply $2$ by $1$ .

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

Step 5.3

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

$x = - 2 \pm 2 i \sqrt{2}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 12}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot 12$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 12}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $12$ .

$x = \frac{- 4 \pm \sqrt{16 - 48}}{2 \cdot 1}$

Step 6.1.3

Subtract $48$ from $16$ .

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

Step 6.1.4

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

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

Step 6.1.5

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

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

Step 6.1.6

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

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

Step 6.1.7

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

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Step 6.1.7.1

Factor $16$ out of $32$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (2)}}{2 \cdot 1}$

Step 6.1.7.2

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

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

Step 6.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (4 \sqrt{2})}{2 \cdot 1}$

Step 6.1.9

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

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

Step 6.2

Multiply $2$ by $1$ .

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

Step 6.3

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

$x = - 2 \pm 2 i \sqrt{2}$

Step 6.4

Change the $\pm$ to $+$ .

$x = - 2 + 2 i \sqrt{2}$

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 $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 12}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 12$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 12}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $12$ .

$x = \frac{- 4 \pm \sqrt{16 - 48}}{2 \cdot 1}$

Step 7.1.3

Subtract $48$ from $16$ .

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

Step 7.1.4

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

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

Step 7.1.5

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

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

Step 7.1.6

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

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

Step 7.1.7

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

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Step 7.1.7.1

Factor $16$ out of $32$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (2)}}{2 \cdot 1}$

Step 7.1.7.2

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

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

Step 7.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (4 \sqrt{2})}{2 \cdot 1}$

Step 7.1.9

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

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

Step 7.2

Multiply $2$ by $1$ .

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

Step 7.3

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

$x = - 2 \pm 2 i \sqrt{2}$

Step 7.4

Change the $\pm$ to $-$ .

$x = - 2 - 2 i \sqrt{2}$

Step 8

The final answer is the combination of both solutions.

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