Solve Using the Quadratic Formula 2/3w^2+3/4w=-1/6

$\frac{2}{3} w^{2} + \frac{3}{4} w = - \frac{1}{6}$

Step 1

Move all terms to the left side of the equation and simplify.

Simplify the left side.

Simplify each term.

Combine $\frac{2}{3}$ and $w^{2}$ .

$\frac{2 w^{2}}{3} + \frac{3}{4} w = - \frac{1}{6}$

Combine $\frac{3}{4}$ and $w$ .

$\frac{2 w^{2}}{3} + \frac{3 w}{4} = - \frac{1}{6}$

Add $\frac{1}{6}$ to both sides of the equation.

$\frac{2 w^{2}}{3} + \frac{3 w}{4} + \frac{1}{6} = 0$

Step 2

Multiply through by the least common denominator $12$ , then simplify.

Apply the distributive property.

$12 (\frac{2 w^{2}}{3}) + 12 (\frac{3 w}{4}) + 12 (\frac{1}{6}) = 0$

Simplify.

Cancel the common factor of $3$ .

Factor $3$ out of $12$ .

$3 (4) (\frac{2 w^{2}}{3}) + 12 (\frac{3 w}{4}) + 12 (\frac{1}{6}) = 0$

Cancel the common factor.

$3 \cdot (4 (\frac{2 w^{2}}{3})) + 12 (\frac{3 w}{4}) + 12 (\frac{1}{6}) = 0$

Rewrite the expression.

$4 (2 w^{2}) + 12 (\frac{3 w}{4}) + 12 (\frac{1}{6}) = 0$

Multiply $2$ by $4$ .

$8 w^{2} + 12 (\frac{3 w}{4}) + 12 (\frac{1}{6}) = 0$

Cancel the common factor of $4$ .

Factor $4$ out of $12$ .

$8 w^{2} + 4 (3) (\frac{3 w}{4}) + 12 (\frac{1}{6}) = 0$

Cancel the common factor.

$8 w^{2} + 4 \cdot (3 (\frac{3 w}{4})) + 12 (\frac{1}{6}) = 0$

Rewrite the expression.

$8 w^{2} + 3 (3 w) + 12 (\frac{1}{6}) = 0$

Multiply $3$ by $3$ .

$8 w^{2} + 9 w + 12 (\frac{1}{6}) = 0$

Cancel the common factor of $6$ .

Factor $6$ out of $12$ .

$8 w^{2} + 9 w + 6 (2) (\frac{1}{6}) = 0$

Cancel the common factor.

$8 w^{2} + 9 w + 6 \cdot (2 (\frac{1}{6})) = 0$

Rewrite the expression.

$8 w^{2} + 9 w + 2 = 0$

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 $a = 8$ , $b = 9$ , and $c = 2$ into the quadratic formula and solve for $w$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (8 \cdot 2)}}{2 \cdot 8}$

Step 5

Simplify.

Simplify the numerator.

Raise $9$ to the power of $2$ .

$w = \frac{- 9 \pm \sqrt{81 - 4 \cdot 8 \cdot 2}}{2 \cdot 8}$

Multiply $- 4 \cdot 8 \cdot 2$ .

Multiply $- 4$ by $8$ .

$w = \frac{- 9 \pm \sqrt{81 - 32 \cdot 2}}{2 \cdot 8}$

Multiply $- 32$ by $2$ .

$w = \frac{- 9 \pm \sqrt{81 - 64}}{2 \cdot 8}$

Subtract $64$ from $81$ .

$w = \frac{- 9 \pm \sqrt{17}}{2 \cdot 8}$

Multiply $2$ by $8$ .

$w = \frac{- 9 \pm \sqrt{17}}{16}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$w = \frac{- 9 \pm \sqrt{17}}{16}$

Decimal Form:

$w = - 0.30480589 \dots, - 0.82019410 \dots$

Solve Using the Quadratic Formula 2/3w^2+3/4w=-1/6

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