Question

Using the quadratic formula, find the solution to 3x^2 + 18x + 24 = 0

Answer 1

To solve the quadratic equation \(3x^2 + 18x + 24 = 0\) using the quadratic formula, we first identify the coefficients:

- \(a = 3\)
- \(b = 18\)
- \(c = 24\)

The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Next, we calculate the discriminant \(b^2 - 4ac\):

\[
b^2 = 18^2 = 324
\]
\[
4ac = 4 \cdot 3 \cdot 24 = 288
\]
\[
b^2 - 4ac = 324 - 288 = 36
\]

Now that we have the discriminant, we can substitute it back into the quadratic formula. First, we find the square root of the discriminant:

\[
\sqrt{36} = 6
\]

Now we can plug in the values into the formula:

\[
x = \frac{-18 \pm 6}{2 \cdot 3} = \frac{-18 \pm 6}{6}
\]

This gives us two possible solutions:

1. When we use the plus sign:

\[
x = \frac{-18 + 6}{6} = \frac{-12}{6} = -2
\]

2. When we use the minus sign:

\[
x = \frac{-18 - 6}{6} = \frac{-24}{6} = -4
\]

Thus, the solutions to the equation \(3x^2 + 18x + 24 = 0\) are:

\[
x = -2 \quad \text{and} \quad x = -4
\]