Precalculus Examples

$9 x^{2} + 24 x + 32 = 0$

Step 1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2

Substitute the values $a = 9$ , $b = 24$ , and $c = 32$ into the quadratic formula and solve for $x$ .

$\frac{- 24 \pm \sqrt{24^{2} - 4 \cdot (9 \cdot 32)}}{2 \cdot 9}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

Raise $24$ to the power of $2$ .

$x = \frac{- 24 \pm \sqrt{576 - 4 \cdot 9 \cdot 32}}{2 \cdot 9}$

Step 3.1.2

Multiply $- 4 \cdot 9 \cdot 32$ .

Tap for more steps...

Step 3.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{- 24 \pm \sqrt{576 - 36 \cdot 32}}{2 \cdot 9}$

Step 3.1.2.2

Multiply $- 36$ by $32$ .

$x = \frac{- 24 \pm \sqrt{576 - 1152}}{2 \cdot 9}$

Step 3.1.3

Subtract $1152$ from $576$ .

$x = \frac{- 24 \pm \sqrt{- 576}}{2 \cdot 9}$

Step 3.1.4

Rewrite $- 576$ as $- 1 (576)$ .

$x = \frac{- 24 \pm \sqrt{- 1 \cdot 576}}{2 \cdot 9}$

Step 3.1.5

Rewrite $\sqrt{- 1 (576)}$ as $\sqrt{- 1} \cdot \sqrt{576}$ .

$x = \frac{- 24 \pm \sqrt{- 1} \cdot \sqrt{576}}{2 \cdot 9}$

Step 3.1.6

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

$x = \frac{- 24 \pm i \cdot \sqrt{576}}{2 \cdot 9}$

Step 3.1.7

Rewrite $576$ as $24^{2}$ .

$x = \frac{- 24 \pm i \cdot \sqrt{24^{2}}}{2 \cdot 9}$

Step 3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 24 \pm i \cdot 24}{2 \cdot 9}$

Step 3.1.9

Move $24$ to the left of $i$ .

$x = \frac{- 24 \pm 24 i}{2 \cdot 9}$

Step 3.2

Multiply $2$ by $9$ .

$x = \frac{- 24 \pm 24 i}{18}$

Step 3.3

Simplify $\frac{- 24 \pm 24 i}{18}$ .

$x = \frac{- 4 \pm 4 i}{3}$