Solve for w 4w^2+19w+18=0

$4 w^{2} + 19 w + 18 = 0$

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 $a = 4$ , $b = 19$ , and $c = 18$ into the quadratic formula and solve for $w$ .

$\frac{- 19 \pm \sqrt{19^{2} - 4 \cdot (4 \cdot 18)}}{2 \cdot 4}$

Step 3

Simplify.

Simplify the numerator.

Raise $19$ to the power of $2$ .

$w = \frac{- 19 \pm \sqrt{361 - 4 \cdot 4 \cdot 18}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot 18$ .

Multiply $- 4$ by $4$ .

$w = \frac{- 19 \pm \sqrt{361 - 16 \cdot 18}}{2 \cdot 4}$

Multiply $- 16$ by $18$ .

$w = \frac{- 19 \pm \sqrt{361 - 288}}{2 \cdot 4}$

Subtract $288$ from $361$ .

$w = \frac{- 19 \pm \sqrt{73}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$w = \frac{- 19 \pm \sqrt{73}}{8}$

Step 4

The final answer is the combination of both solutions.

$w = - \frac{19 - \sqrt{73}}{8}, - \frac{19 + \sqrt{73}}{8}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$w = - \frac{19 - \sqrt{73}}{8}, - \frac{19 + \sqrt{73}}{8}$

Decimal Form:

$w = - 1.30699953 \dots, - 3.44300046 \dots$