To solve the quadratic equation \(3x^2 + 18x + 24 = 0\) using the quadratic formula, we start by identifying the coefficients:
- \(a = 3\)
- \(b = 18\)
- \(c = 24\)
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Now, we need to 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, we can plug the values into the quadratic formula:
\[ x = \frac{-18 \pm \sqrt{36}}{2 \cdot 3} \]
Calculating the square root of the discriminant:
\[ \sqrt{36} = 6 \]
Now substituting this back into the formula:
\[ x = \frac{-18 \pm 6}{6} \]
This gives us two potential solutions:
- When we use the positive root:
\[ x = \frac{-18 + 6}{6} = \frac{-12}{6} = -2 \]
- When we use the negative root:
\[ x = \frac{-18 - 6}{6} = \frac{-24}{6} = -4 \]
Thus, the solutions to the equation \(3x^2 + 18x + 24 = 0\) are:
\[ x = -2 \quad \text{and} \quad x = -4 \]
Therefore, the correct answer from the choices given is:
x = −2, x = −4.