Question

solve for x.
$4x^2 = -12x - 5$
if there is more than one solution, separate them with commas.
if there is no solution, click on
o solution\.
x =
(there are buttons for fractions, mixed numbers, no solution, multiple solutions, square root, and clear/delete buttons)

Explanation:

Step1: Rearrange the equation

First, we need to rewrite the given equation \(4x^{2}=-12x - 5\) in standard quadratic form \(ax^{2}+bx + c = 0\).
Add \(12x\) and \(5\) to both sides of the equation:
\(4x^{2}+12x + 5=0\)

Step2: Use the quadratic formula

The quadratic formula for a quadratic equation \(ax^{2}+bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
For the equation \(4x^{2}+12x + 5 = 0\), we have \(a = 4\), \(b = 12\), and \(c = 5\).
First, calculate the discriminant \(\Delta=b^{2}-4ac\).
Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula:
\(\Delta=(12)^{2}-4\times4\times5\)
\(=144 - 80\)
\(= 64\)

Then, substitute \(a\), \(b\), and \(\Delta\) into the quadratic formula:
\(x=\frac{-12\pm\sqrt{64}}{2\times4}\)
Since \(\sqrt{64}=8\), we have:
\(x=\frac{-12\pm8}{8}\)

Step3: Find the two solutions

We have two cases for the \(\pm\) sign:

• Case 1: When we take the plus sign

\(x=\frac{-12 + 8}{8}=\frac{-4}{8}=-\frac{1}{2}\)

• Case 2: When we take the minus sign

\(x=\frac{-12-8}{8}=\frac{-20}{8}=-\frac{5}{2}\)

Answer:

\(-\frac{1}{2},-\frac{5}{2}\)