Question

solve for w.

4w² + 12w = -9

if there is more than one solution, separate them with commas.
if there is no solution, click on
o solution\.

w =

there is a box for input and a calculator - like interface with buttons including no solution, square root, etc.

Explanation:

Step1: Rewrite the equation

First, we rewrite the given equation \(4w^{2}+12w = - 9\) in standard quadratic form \(ax^{2}+bx + c=0\) (here \(x = w\)). Add 9 to both sides of the equation:
\(4w^{2}+12w + 9=0\)

Step2: Identify coefficients

For the quadratic equation \(ax^{2}+bx + c = 0\), in our case \(a = 4\), \(b=12\), and \(c = 9\).

Step3: Use the quadratic formula or factoring

We can try factoring. Notice that \(4w^{2}+12w + 9=(2w + 3)^{2}\) (since \((a + b)^{2}=a^{2}+2ab + b^{2}\), here \(a = 2w\), \(b = 3\), \(a^{2}=4w^{2}\), \(2ab=2\times(2w)\times3 = 12w\), \(b^{2}=9\)). So we have \((2w + 3)^{2}=0\)

Step4: Solve for w

Set \(2w+3 = 0\). Subtract 3 from both sides: \(2w=-3\). Then divide both sides by 2: \(w=-\frac{3}{2}\)

Answer:

\(-\frac{3}{2}\)