Solve for u 4u^2+9=-12u

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

Step 1

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

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

Step 2

Factor using the perfect square rule.

Rearrange terms.

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

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

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

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

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

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

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

Rewrite the polynomial.

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

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

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

Step 3

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

$2 u + 3 = 0$

Step 4

Solve for $u$ .

Subtract $3$ from both sides of the equation.

$2 u = - 3$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $u$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$u = - 1.5$

Mixed Number Form:

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