Algebra Examples

$3 x^{2} + 4 x = - 5$

Step 1

Add $5$ to both sides of the equation.

$3 x^{2} + 4 x + 5 = 0$

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 $a = 3$ , $b = 4$ , and $c = 5$ into the quadratic formula and solve for $x$ .

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

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 $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 3 \cdot 5}}{2 \cdot 3}$

Step 4.1.2

Multiply $- 4 \cdot 3 \cdot 5$ .

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Step 4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 4 \pm \sqrt{16 - 12 \cdot 5}}{2 \cdot 3}$

Step 4.1.2.2

Multiply $- 12$ by $5$ .

$x = \frac{- 4 \pm \sqrt{16 - 60}}{2 \cdot 3}$

Step 4.1.3

Subtract $60$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 44}}{2 \cdot 3}$

Step 4.1.4

Rewrite $- 44$ as $- 1 (44)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 44}}{2 \cdot 3}$

Step 4.1.5

Rewrite $\sqrt{- 1 (44)}$ as $\sqrt{- 1} \cdot \sqrt{44}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{44}}{2 \cdot 3}$

Step 4.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{44}}{2 \cdot 3}$

Step 4.1.7

Rewrite $44$ as $2^{2} \cdot 11$ .

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Step 4.1.7.1

Factor $4$ out of $44$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4 (11)}}{2 \cdot 3}$

Step 4.1.7.2

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

$x = \frac{- 4 \pm i \cdot \sqrt{2^{2} \cdot 11}}{2 \cdot 3}$

Step 4.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (2 \sqrt{11})}{2 \cdot 3}$

Step 4.1.9

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

$x = \frac{- 4 \pm 2 i \sqrt{11}}{2 \cdot 3}$

Step 4.2

Multiply $2$ by $3$ .

$x = \frac{- 4 \pm 2 i \sqrt{11}}{6}$

Step 4.3

Simplify $\frac{- 4 \pm 2 i \sqrt{11}}{6}$ .

$x = \frac{- 2 \pm i \sqrt{11}}{3}$