Algebra Examples

$3 x^{2} = - 12 - 5 x$

Step 1

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

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Step 1.1

Add $12$ to both sides of the equation.

$3 x^{2} + 12 = - 5 x$

Step 1.2

Add $5 x$ to both sides of the equation.

$3 x^{2} + 12 + 5 x = 0$

Step 2

Reorder terms.

$3 x^{2} + 5 x + 12 = 0$

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 $a = 3$ , $b = 5$ , and $c = 12$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (3 \cdot 12)}}{2 \cdot 3}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 3 \cdot 12}}{2 \cdot 3}$

Step 5.1.2

Multiply $- 4 \cdot 3 \cdot 12$ .

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Step 5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 5 \pm \sqrt{25 - 12 \cdot 12}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $12$ .

$x = \frac{- 5 \pm \sqrt{25 - 144}}{2 \cdot 3}$

Step 5.1.3

Subtract $144$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 119}}{2 \cdot 3}$

Step 5.1.4

Rewrite $- 119$ as $- 1 (119)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 119}}{2 \cdot 3}$

Step 5.1.5

Rewrite $\sqrt{- 1 (119)}$ as $\sqrt{- 1} \cdot \sqrt{119}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{119}}{2 \cdot 3}$

Step 5.1.6

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

$x = \frac{- 5 \pm i \sqrt{119}}{2 \cdot 3}$

Step 5.2

Multiply $2$ by $3$ .

$x = \frac{- 5 \pm i \sqrt{119}}{6}$

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 $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 3 \cdot 12}}{2 \cdot 3}$

Step 6.1.2

Multiply $- 4 \cdot 3 \cdot 12$ .

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Step 6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 5 \pm \sqrt{25 - 12 \cdot 12}}{2 \cdot 3}$

Step 6.1.2.2

Multiply $- 12$ by $12$ .

$x = \frac{- 5 \pm \sqrt{25 - 144}}{2 \cdot 3}$

Step 6.1.3

Subtract $144$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 119}}{2 \cdot 3}$

Step 6.1.4

Rewrite $- 119$ as $- 1 (119)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 119}}{2 \cdot 3}$

Step 6.1.5

Rewrite $\sqrt{- 1 (119)}$ as $\sqrt{- 1} \cdot \sqrt{119}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{119}}{2 \cdot 3}$

Step 6.1.6

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

$x = \frac{- 5 \pm i \sqrt{119}}{2 \cdot 3}$

Step 6.2

Multiply $2$ by $3$ .

$x = \frac{- 5 \pm i \sqrt{119}}{6}$

Step 6.3

Change the $\pm$ to $+$ .

$x = \frac{- 5 + i \sqrt{119}}{6}$

Step 6.4

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 + i \sqrt{119}}{6}$

Step 6.5

Factor $- 1$ out of $i \sqrt{119}$ .

$x = \frac{- 1 \cdot 5 - (- i \sqrt{119})}{6}$

Step 6.6

Factor $- 1$ out of $- 1 (5) - (- i \sqrt{119})$ .

$x = \frac{- 1 (5 - i \sqrt{119})}{6}$

Step 6.7

Move the negative in front of the fraction.

$x = - \frac{5 - i \sqrt{119}}{6}$

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 $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 3 \cdot 12}}{2 \cdot 3}$

Step 7.1.2

Multiply $- 4 \cdot 3 \cdot 12$ .

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Step 7.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 5 \pm \sqrt{25 - 12 \cdot 12}}{2 \cdot 3}$

Step 7.1.2.2

Multiply $- 12$ by $12$ .

$x = \frac{- 5 \pm \sqrt{25 - 144}}{2 \cdot 3}$

Step 7.1.3

Subtract $144$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 119}}{2 \cdot 3}$

Step 7.1.4

Rewrite $- 119$ as $- 1 (119)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 119}}{2 \cdot 3}$

Step 7.1.5

Rewrite $\sqrt{- 1 (119)}$ as $\sqrt{- 1} \cdot \sqrt{119}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{119}}{2 \cdot 3}$

Step 7.1.6

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

$x = \frac{- 5 \pm i \sqrt{119}}{2 \cdot 3}$

Step 7.2

Multiply $2$ by $3$ .

$x = \frac{- 5 \pm i \sqrt{119}}{6}$

Step 7.3

Change the $\pm$ to $-$ .

$x = \frac{- 5 - i \sqrt{119}}{6}$

Step 7.4

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 - i \sqrt{119}}{6}$

Step 7.5

Factor $- 1$ out of $- i \sqrt{119}$ .

$x = \frac{- 1 \cdot 5 - (i \sqrt{119})}{6}$

Step 7.6

Factor $- 1$ out of $- 1 (5) - (i \sqrt{119})$ .

$x = \frac{- 1 (5 + i \sqrt{119})}{6}$

Step 7.7

Move the negative in front of the fraction.

$x = - \frac{5 + i \sqrt{119}}{6}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{5 - i \sqrt{119}}{6}, - \frac{5 + i \sqrt{119}}{6}$