Pre-Algebra Examples

$5 x^{2} + 2 x + 10 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 5$ , $b = 2$ , and $c = 10$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (5 \cdot 10)}}{2 \cdot 5}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 5 \cdot 10}}{2 \cdot 5}$

Step 3.1.2

Multiply $- 4 \cdot 5 \cdot 10$ .

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Step 3.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 2 \pm \sqrt{4 - 20 \cdot 10}}{2 \cdot 5}$

Step 3.1.2.2

Multiply $- 20$ by $10$ .

$x = \frac{- 2 \pm \sqrt{4 - 200}}{2 \cdot 5}$

Step 3.1.3

Subtract $200$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 196}}{2 \cdot 5}$

Step 3.1.4

Rewrite $- 196$ as $- 1 (196)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 196}}{2 \cdot 5}$

Step 3.1.5

Rewrite $\sqrt{- 1 (196)}$ as $\sqrt{- 1} \cdot \sqrt{196}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{196}}{2 \cdot 5}$

Step 3.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{196}}{2 \cdot 5}$

Step 3.1.7

Rewrite $196$ as $14^{2}$ .

$x = \frac{- 2 \pm i \cdot \sqrt{14^{2}}}{2 \cdot 5}$

Step 3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 2 \pm i \cdot 14}{2 \cdot 5}$

Step 3.1.9

Move $14$ to the left of $i$ .

$x = \frac{- 2 \pm 14 i}{2 \cdot 5}$

Step 3.2

Multiply $2$ by $5$ .

$x = \frac{- 2 \pm 14 i}{10}$

Step 3.3

Simplify $\frac{- 2 \pm 14 i}{10}$ .

$x = \frac{- 1 \pm 7 i}{5}$

Step 4

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

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 5 \cdot 10}}{2 \cdot 5}$

Step 4.1.2

Multiply $- 4 \cdot 5 \cdot 10$ .

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Step 4.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 2 \pm \sqrt{4 - 20 \cdot 10}}{2 \cdot 5}$

Step 4.1.2.2

Multiply $- 20$ by $10$ .

$x = \frac{- 2 \pm \sqrt{4 - 200}}{2 \cdot 5}$

Step 4.1.3

Subtract $200$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 196}}{2 \cdot 5}$

Step 4.1.4

Rewrite $- 196$ as $- 1 (196)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 196}}{2 \cdot 5}$

Step 4.1.5

Rewrite $\sqrt{- 1 (196)}$ as $\sqrt{- 1} \cdot \sqrt{196}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{196}}{2 \cdot 5}$

Step 4.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{196}}{2 \cdot 5}$

Step 4.1.7

Rewrite $196$ as $14^{2}$ .

$x = \frac{- 2 \pm i \cdot \sqrt{14^{2}}}{2 \cdot 5}$

Step 4.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 2 \pm i \cdot 14}{2 \cdot 5}$

Step 4.1.9

Move $14$ to the left of $i$ .

$x = \frac{- 2 \pm 14 i}{2 \cdot 5}$

Step 4.2

Multiply $2$ by $5$ .

$x = \frac{- 2 \pm 14 i}{10}$

Step 4.3

Simplify $\frac{- 2 \pm 14 i}{10}$ .

$x = \frac{- 1 \pm 7 i}{5}$

Step 4.4

Change the $\pm$ to $+$ .

$x = \frac{- 1 + 7 i}{5}$

Step 4.5

Split the fraction $\frac{- 1 + 7 i}{5}$ into two fractions.

$x = \frac{- 1}{5} + \frac{7 i}{5}$

Step 4.6

Move the negative in front of the fraction.

$x = - \frac{1}{5} + \frac{7 i}{5}$

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 $2$ to the power of $2$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 5 \cdot 10}}{2 \cdot 5}$

Step 5.1.2

Multiply $- 4 \cdot 5 \cdot 10$ .

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Step 5.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 2 \pm \sqrt{4 - 20 \cdot 10}}{2 \cdot 5}$

Step 5.1.2.2

Multiply $- 20$ by $10$ .

$x = \frac{- 2 \pm \sqrt{4 - 200}}{2 \cdot 5}$

Step 5.1.3

Subtract $200$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 196}}{2 \cdot 5}$

Step 5.1.4

Rewrite $- 196$ as $- 1 (196)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 196}}{2 \cdot 5}$

Step 5.1.5

Rewrite $\sqrt{- 1 (196)}$ as $\sqrt{- 1} \cdot \sqrt{196}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{196}}{2 \cdot 5}$

Step 5.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{196}}{2 \cdot 5}$

Step 5.1.7

Rewrite $196$ as $14^{2}$ .

$x = \frac{- 2 \pm i \cdot \sqrt{14^{2}}}{2 \cdot 5}$

Step 5.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 2 \pm i \cdot 14}{2 \cdot 5}$

Step 5.1.9

Move $14$ to the left of $i$ .

$x = \frac{- 2 \pm 14 i}{2 \cdot 5}$

Step 5.2

Multiply $2$ by $5$ .

$x = \frac{- 2 \pm 14 i}{10}$

Step 5.3

Simplify $\frac{- 2 \pm 14 i}{10}$ .

$x = \frac{- 1 \pm 7 i}{5}$

Step 5.4

Change the $\pm$ to $-$ .

$x = \frac{- 1 - 7 i}{5}$

Step 5.5

Split the fraction $\frac{- 1 - 7 i}{5}$ into two fractions.

$x = \frac{- 1}{5} + \frac{- 7 i}{5}$

Step 5.6

Simplify each term.

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Step 5.6.1

Move the negative in front of the fraction.

$x = - \frac{1}{5} + \frac{- 7 i}{5}$

Step 5.6.2

Move the negative in front of the fraction.

$x = - \frac{1}{5} - \frac{7 i}{5}$

Step 6

The final answer is the combination of both solutions.

$x = - \frac{1}{5} + \frac{7 i}{5}, - \frac{1}{5} - \frac{7 i}{5}$