To solve the quadratic equation [tex]\(3x^2 + 24x - 24 = 0\)[/tex], let's follow the steps to find the roots step-by-step.
### Step 1: Check that the equation is in standard form
The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex]. In this problem, we have:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = 24\)[/tex]
- [tex]\(c = -24\)[/tex]
### Step 2: Use the quadratic formula
The quadratic formula for [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
### Step 3: Identify the coefficients
Given:
[tex]\[a = 3\][/tex]
[tex]\[b = 24\][/tex]
[tex]\[c = -24\][/tex]
### Step 4: Calculate the discriminant [tex]\(\Delta\)[/tex]
The discriminant [tex]\(\Delta\)[/tex] is found using the formula [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
[tex]\[
\Delta = 24^2 - 4 \cdot 3 \cdot (-24)
\][/tex]
[tex]\[
\Delta = 576 + 288
\][/tex]
[tex]\[
\Delta = 864
\][/tex]
### Step 5: Plug the discriminant back into the quadratic formula
Now, substituting [tex]\(\Delta\)[/tex] and the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex] into the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
[tex]\[
x = \frac{-24 \pm \sqrt{864}}{2 \cdot 3}
\][/tex]
### Step 6: Simplify under the square root
Recognize that:
[tex]\[
\sqrt{864} = \sqrt{144 \cdot 6} = 12\sqrt{6}
\][/tex]
### Step 7: Substitute and simplify
Substitute [tex]\(\sqrt{864} = 12\sqrt{6}\)[/tex] back into the quadratic formula:
[tex]\[
x = \frac{-24 \pm 12\sqrt{6}}{6}
\][/tex]
Now, split the two terms:
[tex]\[
x = \frac{-24}{6} \pm \frac{12\sqrt{6}}{6}
\][/tex]
[tex]\[
x = -4 \pm 2\sqrt{6}
\][/tex]
### Step 8: State the final result
Hence, the solutions to the quadratic equation [tex]\(3x^2 + 24x - 24 = 0\)[/tex] are:
[tex]\[
x = -4 \pm 2\sqrt{6}
\][/tex]
Thus, the correct answer choice is:
A. [tex]\(x = -4 \pm 2\sqrt{6}\)[/tex]
