Algebra Examples

$5 - 2 (3 x^{2} - 2 x) = x^{2} + 7 - 2 x^{2}$

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 each term.

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Step 1.1.1.1

Apply the distributive property.

$5 - 2 (3 x^{2}) - 2 (- 2 x) = x^{2} + 7 - 2 x^{2}$

Step 1.1.1.2

Multiply $3$ by $- 2$ .

$5 - 6 x^{2} - 2 (- 2 x) = x^{2} + 7 - 2 x^{2}$

Step 1.1.1.3

Multiply $- 2$ by $- 2$ .

$5 - 6 x^{2} + 4 x = x^{2} + 7 - 2 x^{2}$

Step 1.2

Simplify the right side.

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Step 1.2.1

Subtract $2 x^{2}$ from $x^{2}$ .

$5 - 6 x^{2} + 4 x = - x^{2} + 7$

Step 1.3

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

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Step 1.3.1

Add $x^{2}$ to both sides of the equation.

$5 - 6 x^{2} + 4 x + x^{2} = 7$

Step 1.3.2

Subtract $7$ from both sides of the equation.

$5 - 6 x^{2} + 4 x + x^{2} - 7 = 0$

Step 1.4

Simplify $5 - 6 x^{2} + 4 x + x^{2} - 7$ .

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Step 1.4.1

Subtract $7$ from $5$ .

$- 6 x^{2} + 4 x + x^{2} - 2 = 0$

Step 1.4.2

Add $- 6 x^{2}$ and $x^{2}$ .

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

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

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

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 $2$ .

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

Step 4.1.2

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

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Step 4.1.2.1

Multiply $- 4$ by $- 5$ .

$x = \frac{- 4 \pm \sqrt{16 + 20 \cdot - 2}}{2 \cdot - 5}$

Step 4.1.2.2

Multiply $20$ by $- 2$ .

$x = \frac{- 4 \pm \sqrt{16 - 40}}{2 \cdot - 5}$

Step 4.1.3

Subtract $40$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 24}}{2 \cdot - 5}$

Step 4.1.4

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

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 24}}{2 \cdot - 5}$

Step 4.1.5

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

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{24}}{2 \cdot - 5}$

Step 4.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{24}}{2 \cdot - 5}$

Step 4.1.7

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

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Step 4.1.7.1

Factor $4$ out of $24$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4 (6)}}{2 \cdot - 5}$

Step 4.1.7.2

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

$x = \frac{- 4 \pm i \cdot \sqrt{2^{2} \cdot 6}}{2 \cdot - 5}$

Step 4.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (2 \sqrt{6})}{2 \cdot - 5}$

Step 4.1.9

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

$x = \frac{- 4 \pm 2 i \sqrt{6}}{2 \cdot - 5}$

Step 4.2

Multiply $2$ by $- 5$ .

$x = \frac{- 4 \pm 2 i \sqrt{6}}{- 10}$

Step 4.3

Simplify $\frac{- 4 \pm 2 i \sqrt{6}}{- 10}$ .

$x = \frac{2 \pm i \sqrt{6}}{5}$