Algebra Examples

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

Step 1

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

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

Step 2

Add $12$ to both sides of the equation.

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

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

The final answer is the combination of both solutions.

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