Pre-Algebra Examples

$2 x^{2} + 4 x = - 5$

Step 1

Add $5$ to both sides of the equation.

$2 x^{2} + 4 x + 5 = 0$

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 $a = 2$ , $b = 4$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (2 \cdot 5)}}{2 \cdot 2}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 2 \cdot 5}}{2 \cdot 2}$

Step 4.1.2

Multiply $- 4 \cdot 2 \cdot 5$ .

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 8 \cdot 5}}{2 \cdot 2}$

Step 4.1.2.2

Multiply $- 8$ by $5$ .

$x = \frac{- 4 \pm \sqrt{16 - 40}}{2 \cdot 2}$

Step 4.1.3

Subtract $40$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 24}}{2 \cdot 2}$

Step 4.1.4

Rewrite $- 24$ as $- 1 (24)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 24}}{2 \cdot 2}$

Step 4.1.5

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{24}}{2 \cdot 2}$

Step 4.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{24}}{2 \cdot 2}$

Step 4.1.7

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 4.1.7.1

Factor $4$ out of $24$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4 (6)}}{2 \cdot 2}$

Step 4.1.7.2

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

$x = \frac{- 4 \pm i \cdot \sqrt{2^{2} \cdot 6}}{2 \cdot 2}$

Step 4.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (2 \sqrt{6})}{2 \cdot 2}$

Step 4.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 4 \pm 2 i \sqrt{6}}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$x = \frac{- 4 \pm 2 i \sqrt{6}}{4}$

Step 4.3

Simplify $\frac{- 4 \pm 2 i \sqrt{6}}{4}$ .

$x = \frac{- 2 \pm i \sqrt{6}}{2}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 2 \cdot 5}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot 5$ .

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 8 \cdot 5}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $5$ .

$x = \frac{- 4 \pm \sqrt{16 - 40}}{2 \cdot 2}$

Step 5.1.3

Subtract $40$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 24}}{2 \cdot 2}$

Step 5.1.4

Rewrite $- 24$ as $- 1 (24)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 24}}{2 \cdot 2}$

Step 5.1.5

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{24}}{2 \cdot 2}$

Step 5.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{24}}{2 \cdot 2}$

Step 5.1.7

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 5.1.7.1

Factor $4$ out of $24$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4 (6)}}{2 \cdot 2}$

Step 5.1.7.2

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

$x = \frac{- 4 \pm i \cdot \sqrt{2^{2} \cdot 6}}{2 \cdot 2}$

Step 5.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (2 \sqrt{6})}{2 \cdot 2}$

Step 5.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 4 \pm 2 i \sqrt{6}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$x = \frac{- 4 \pm 2 i \sqrt{6}}{4}$

Step 5.3

Simplify $\frac{- 4 \pm 2 i \sqrt{6}}{4}$ .

$x = \frac{- 2 \pm i \sqrt{6}}{2}$

Step 5.4

Change the $\pm$ to $+$ .

$x = \frac{- 2 + i \sqrt{6}}{2}$

Step 5.5

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{- 1 \cdot 2 + i \sqrt{6}}{2}$

Step 5.6

Factor $- 1$ out of $i \sqrt{6}$ .

$x = \frac{- 1 \cdot 2 - (- i \sqrt{6})}{2}$

Step 5.7

Factor $- 1$ out of $- 1 (2) - (- i \sqrt{6})$ .

$x = \frac{- 1 (2 - i \sqrt{6})}{2}$

Step 5.8

Move the negative in front of the fraction.

$x = - \frac{2 - i \sqrt{6}}{2}$

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 $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 2 \cdot 5}}{2 \cdot 2}$

Step 6.1.2

Multiply $- 4 \cdot 2 \cdot 5$ .

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Step 6.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 8 \cdot 5}}{2 \cdot 2}$

Step 6.1.2.2

Multiply $- 8$ by $5$ .

$x = \frac{- 4 \pm \sqrt{16 - 40}}{2 \cdot 2}$

Step 6.1.3

Subtract $40$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 24}}{2 \cdot 2}$

Step 6.1.4

Rewrite $- 24$ as $- 1 (24)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 24}}{2 \cdot 2}$

Step 6.1.5

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{24}}{2 \cdot 2}$

Step 6.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{24}}{2 \cdot 2}$

Step 6.1.7

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 6.1.7.1

Factor $4$ out of $24$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4 (6)}}{2 \cdot 2}$

Step 6.1.7.2

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

$x = \frac{- 4 \pm i \cdot \sqrt{2^{2} \cdot 6}}{2 \cdot 2}$

Step 6.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (2 \sqrt{6})}{2 \cdot 2}$

Step 6.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 4 \pm 2 i \sqrt{6}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$x = \frac{- 4 \pm 2 i \sqrt{6}}{4}$

Step 6.3

Simplify $\frac{- 4 \pm 2 i \sqrt{6}}{4}$ .

$x = \frac{- 2 \pm i \sqrt{6}}{2}$

Step 6.4

Change the $\pm$ to $-$ .

$x = \frac{- 2 - i \sqrt{6}}{2}$

Step 6.5

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{- 1 \cdot 2 - i \sqrt{6}}{2}$

Step 6.6

Factor $- 1$ out of $- i \sqrt{6}$ .

$x = \frac{- 1 \cdot 2 - (i \sqrt{6})}{2}$

Step 6.7

Factor $- 1$ out of $- 1 (2) - (i \sqrt{6})$ .

$x = \frac{- 1 (2 + i \sqrt{6})}{2}$

Step 6.8

Move the negative in front of the fraction.

$x = - \frac{2 + i \sqrt{6}}{2}$

Step 7

The final answer is the combination of both solutions.

$x = - \frac{2 - i \sqrt{6}}{2}, - \frac{2 + i \sqrt{6}}{2}$