Solve for w 4w^2=-12w-9

$4 w^{2} = - 12 w - 9$

Step 1

Add $12 w$ to both sides of the equation.

$4 w^{2} + 12 w = - 9$

Step 2

Add $9$ to both sides of the equation.

$4 w^{2} + 12 w + 9 = 0$

Step 3

Factor using the perfect square rule.

Rewrite $4 w^{2}$ as ${(2 w)}^{2}$ .

${(2 w)}^{2} + 12 w + 9 = 0$

Rewrite $9$ as $3^{2}$ .

${(2 w)}^{2} + 12 w + 3^{2} = 0$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$12 w = 2 \cdot (2 w) \cdot 3$

Rewrite the polynomial.

${(2 w)}^{2} + 2 \cdot (2 w) \cdot 3 + 3^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 2 w$ and $b = 3$ .

${(2 w + 3)}^{2} = 0$

Step 4

Set the $2 w + 3$ equal to $0$ .

$2 w + 3 = 0$

Step 5

Solve for $w$ .

Subtract $3$ from both sides of the equation.

$2 w = - 3$

Divide each term in $2 w = - 3$ by $2$ and simplify.

Divide each term in $2 w = - 3$ by $2$ .

$\frac{2 w}{2} = \frac{- 3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 w}{2} = \frac{- 3}{2}$

Divide $w$ by $1$ .

$w = \frac{- 3}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$w = - \frac{3}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$w = - \frac{3}{2}$

Decimal Form:

$w = - 1.5$

Mixed Number Form:

$w = - 1 \frac{1}{2}$

Solve for w 4w^2=-12w-9

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