Algebra Examples

$x (2 x + 2) = - 10$

Step 1

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

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify $x (2 x + 2)$ .

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Step 1.1.1.1

Simplify by multiplying through.

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Step 1.1.1.1.1

Apply the distributive property.

$x (2 x) + x \cdot 2 = - 10$

Step 1.1.1.1.2

Reorder.

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Step 1.1.1.1.2.1

Rewrite using the commutative property of multiplication.

$2 x \cdot x + x \cdot 2 = - 10$

Step 1.1.1.1.2.2

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

$2 x \cdot x + 2 \cdot x = - 10$

Step 1.1.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.1.2.1

Move $x$ .

$2 (x \cdot x) + 2 \cdot x = - 10$

Step 1.1.1.2.2

Multiply $x$ by $x$ .

$2 x^{2} + 2 \cdot x = - 10$

$2 x^{2} + 2 x = - 10$

Step 1.2

Add $10$ to both sides of the equation.

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

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

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

Multiply $- 4 \cdot 2 \cdot 10$ .

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 2 \pm \sqrt{4 - 8 \cdot 10}}{2 \cdot 2}$

Step 4.1.2.2

Multiply $- 8$ by $10$ .

$x = \frac{- 2 \pm \sqrt{4 - 80}}{2 \cdot 2}$

Step 4.1.3

Subtract $80$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 76}}{2 \cdot 2}$

Step 4.1.4

Rewrite $- 76$ as $- 1 (76)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 76}}{2 \cdot 2}$

Step 4.1.5

Rewrite $\sqrt{- 1 (76)}$ as $\sqrt{- 1} \cdot \sqrt{76}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{76}}{2 \cdot 2}$

Step 4.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{76}}{2 \cdot 2}$

Step 4.1.7

Rewrite $76$ as $2^{2} \cdot 19$ .

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Step 4.1.7.1

Factor $4$ out of $76$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (19)}}{2 \cdot 2}$

Step 4.1.7.2

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

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 19}}{2 \cdot 2}$

Step 4.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{19})}{2 \cdot 2}$

Step 4.1.9

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

$x = \frac{- 2 \pm 2 i \sqrt{19}}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$x = \frac{- 2 \pm 2 i \sqrt{19}}{4}$

Step 4.3

Simplify $\frac{- 2 \pm 2 i \sqrt{19}}{4}$ .

$x = \frac{- 1 \pm i \sqrt{19}}{2}$