Sure, let's solve the quadratic equation step-by-step:
The given equation is:
[tex]\[ 3x^2 + 24x - 24 = 0. \][/tex]
To solve this quadratic equation, we can use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \][/tex]
where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients of [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constant term respectively in the standard form of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex].
For the given equation [tex]\(3x^2 + 24x - 24 = 0\)[/tex]:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = 24\)[/tex]
- [tex]\(c = -24\)[/tex]
Now, let's plug these values into the quadratic formula:
1. Calculate the discriminant [tex]\(\Delta\)[/tex]:
[tex]\[ \Delta = b^2 - 4ac. \][/tex]
Substituting the values:
[tex]\[ \Delta = 24^2 - 4 \cdot 3 \cdot (-24) \][/tex]
[tex]\[ \Delta = 576 + 288 \][/tex]
[tex]\[ \Delta = 864. \][/tex]
2. Find the square root of the discriminant:
[tex]\[ \sqrt{\Delta} = \sqrt{864}. \][/tex]
Simplifying further:
[tex]\[ \sqrt{864} = \sqrt{144 \times 6} = 12\sqrt{6}. \][/tex]
3. Substitute [tex]\(b\)[/tex], [tex]\(\sqrt{\Delta}\)[/tex], and [tex]\(2a\)[/tex] into the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x = \frac{-24 \pm 12\sqrt{6}}{2 \cdot 3} \][/tex]
[tex]\[ x = \frac{-24 \pm 12\sqrt{6}}{6} \][/tex]
[tex]\[ x = \frac{-24}{6} \pm \frac{12\sqrt{6}}{6} \][/tex]
[tex]\[ x = -4 \pm 2\sqrt{6}. \][/tex]
Thus, the solutions to the equation are:
[tex]\[ x = -4 + 2\sqrt{6} \quad \text{or} \quad x = -4 - 2\sqrt{6}. \][/tex]
Now let's compare these solutions with the given options:
A. [tex]\(x = 2 \pm 4\sqrt{6}\)[/tex]
B. [tex]\(x = -2 \pm 4\sqrt{6}\)[/tex]
C. [tex]\(x = -4 \pm 2\sqrt{6}\)[/tex]
D. [tex]\(x = 4 \pm 2\sqrt{6}\)[/tex]
The correct option is C: [tex]\(x = -4 \pm 2\sqrt{6}\)[/tex].
