To solve the quadratic equation [tex]\( 3x^2 + 24x - 24 = 0 \)[/tex], let's follow a detailed, step-by-step solution.
### Step 1: Identify the coefficients
The given quadratic equation is of the form:
[tex]\[ a x^2 + b x + c = 0 \][/tex]
where [tex]\( a = 3 \)[/tex], [tex]\( b = 24 \)[/tex], and [tex]\( c = -24 \)[/tex].
### Step 2: Use the quadratic formula
The quadratic formula for solving [tex]\( ax^2 + bx + c = 0 \)[/tex] is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
### Step 3: Calculate the discriminant
The discriminant ([tex]\( \Delta \)[/tex]) is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute [tex]\( a = 3 \)[/tex], [tex]\( b = 24 \)[/tex], and [tex]\( c = -24 \)[/tex]:
[tex]\[ \Delta = 24^2 - 4 \cdot 3 \cdot (-24) \][/tex]
[tex]\[ \Delta = 576 + 288 \][/tex]
[tex]\[ \Delta = 864 \][/tex]
### Step 4: Calculate the square root of the discriminant
[tex]\[ \sqrt{\Delta} = \sqrt{864} \][/tex]
[tex]\[ \sqrt{864} = \sqrt{144 \cdot 6} = 12 \sqrt{6} \][/tex]
### Step 5: Apply the quadratic formula
Substitute [tex]\( b = 24 \)[/tex], [tex]\( \sqrt{\Delta} = 12 \sqrt{6} \)[/tex], and [tex]\( a = 3 \)[/tex] into the quadratic formula:
[tex]\[ x = \frac{-24 \pm 12 \sqrt{6}}{6} \][/tex]
### Step 6: Simplify the expression
Separate the fraction:
[tex]\[ x = \frac{-24}{6} \pm \frac{12 \sqrt{6}}{6} \][/tex]
[tex]\[ x = -4 \pm 2 \sqrt{6} \][/tex]
### Step 7: Write the final solutions
The solutions to the quadratic equation [tex]\( 3x^2 + 24x - 24 = 0 \)[/tex] are:
[tex]\[ x = -4 + 2 \sqrt{6} \quad \text{and} \quad x = -4 - 2 \sqrt{6} \][/tex]
Checking the given options:
A. [tex]\( x = 4 \pm 2 \sqrt{6} \)[/tex]
B. [tex]\( x = -2 \pm 4 \sqrt{6} \)[/tex]
C. [tex]\( x = -4 \pm 2 \sqrt{6} \)[/tex]
D. [tex]\( x = 2 \pm 4 \sqrt{6} \)[/tex]
### Step 8: Select the correct answer
Thus, the correct answer is:
C. [tex]\( x = -4 \pm 2 \sqrt{6} \)[/tex]
