To determine which of the following is a solution to the quadratic equation [tex]\(3x^2 + 6x - 24 = 0\)[/tex], we will solve the equation and check which of the given values is a valid solution.
Step-by-step process to solve the quadratic equation [tex]\(3x^2 + 6x - 24 = 0\)[/tex]:
1. Identify the quadratic equation's coefficients:
[tex]\[
ax^2 + bx + c = 0
\][/tex]
Here, the equation is [tex]\(3x^2 + 6x - 24 = 0\)[/tex] with [tex]\(a = 3\)[/tex], [tex]\(b = 6\)[/tex], and [tex]\(c = -24\)[/tex].
2. Use the quadratic formula to find the solutions:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Plug in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 3 \cdot (-24)}}{2 \cdot 3}
\][/tex]
3. Calculate the discriminant:
[tex]\[
b^2 - 4ac = 6^2 - 4 \cdot 3 \cdot (-24) = 36 + 288 = 324
\][/tex]
4. Take the square root of the discriminant:
[tex]\[
\sqrt{324} = 18
\][/tex]
5. Find the two solutions using the quadratic formula:
[tex]\[
x = \frac{-6 + 18}{6} = \frac{12}{6} = 2
\][/tex]
[tex]\[
x = \frac{-6 - 18}{6} = \frac{-24}{6} = -4
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\(3x^2 + 6x - 24 = 0\)[/tex] are [tex]\(x = 2\)[/tex] and [tex]\(x = -4\)[/tex].
Now verify which of the given options matches one of the solutions:
- A. -24
- C. -2
- B. -4
- D. 0
The valid solution from the list is:
- B. -4
So, the correct choice is:
b. B
