Solve Using the Quadratic Formula 4x^2+20x+29=0

$4 x^{2} + 20 x + 29 = 0$

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 $a = 4$ , $b = 20$ , and $c = 29$ into the quadratic formula and solve for $x$ .

$\frac{- 20 \pm \sqrt{20^{2} - 4 \cdot (4 \cdot 29)}}{2 \cdot 4}$

Step 3

Simplify.

Simplify the numerator.

Raise $20$ to the power of $2$ .

$x = \frac{- 20 \pm \sqrt{400 - 4 \cdot 4 \cdot 29}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot 29$ .

Multiply $- 4$ by $4$ .

$x = \frac{- 20 \pm \sqrt{400 - 16 \cdot 29}}{2 \cdot 4}$

Multiply $- 16$ by $29$ .

$x = \frac{- 20 \pm \sqrt{400 - 464}}{2 \cdot 4}$

Subtract $464$ from $400$ .

$x = \frac{- 20 \pm \sqrt{- 64}}{2 \cdot 4}$

Rewrite $- 64$ as $- 1 (64)$ .

$x = \frac{- 20 \pm \sqrt{- 1 \cdot 64}}{2 \cdot 4}$

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$x = \frac{- 20 \pm \sqrt{- 1} \cdot \sqrt{64}}{2 \cdot 4}$

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

$x = \frac{- 20 \pm i \cdot \sqrt{64}}{2 \cdot 4}$

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 20 \pm i \cdot \sqrt{8^{2}}}{2 \cdot 4}$

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

$x = \frac{- 20 \pm i \cdot 8}{2 \cdot 4}$

Move $8$ to the left of $i$ .

$x = \frac{- 20 \pm 8 i}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{- 20 \pm 8 i}{8}$

Simplify $\frac{- 20 \pm 8 i}{8}$ .

$x = \frac{- 5 \pm 2 i}{2}$