Question

solve for g.
$g^2 + 24g + 144 = 0$
write each solution as an integer, proper fraction, or improper fraction in simplest form. if there are multiple solutions, separate them with commas.
$g = $

Explanation:

Step1: Recognize the quadratic form

The equation \(g^2 + 24g + 144 = 0\) is a quadratic equation. We can try to factor it. Notice that it might be a perfect square trinomial. A perfect square trinomial has the form \(a^2+2ab + b^2=(a + b)^2\). Here, \(a = g\), \(2ab=24g\), so \(2b = 24\) which means \(b = 12\), and \(b^2=144\), which matches the constant term. So we can factor the quadratic as \((g + 12)^2=0\).

Step2: Solve for g

To solve \((g + 12)^2=0\), we take the square root of both sides. The square root of a square is the absolute value, but since the right - hand side is 0, we have \(g+12 = 0\). Subtracting 12 from both sides gives \(g=- 12\). Since this is a repeated root (because the quadratic was a perfect square), there is only one distinct solution.

Answer:

\(g=-12\)