Certainly! Let's solve the given equation step-by-step.
We start with the equation:
[tex]\[ 3x^2 - 48 = 0 \][/tex]
### Step 1: Isolate the term with [tex]\(x^2\)[/tex]
To isolate [tex]\(x^2\)[/tex], we first need to add 48 to both sides of the equation:
[tex]\[ 3x^2 - 48 + 48 = 0 + 48 \][/tex]
[tex]\[ 3x^2 = 48 \][/tex]
### Step 2: Solve for [tex]\(x^2\)[/tex]
Next, we divide both sides of the equation by 3 to solve for [tex]\(x^2\)[/tex]:
[tex]\[ \frac{3x^2}{3} = \frac{48}{3} \][/tex]
[tex]\[ x^2 = 16 \][/tex]
### Step 3: Take the square root of both sides
To find [tex]\(x\)[/tex], we take the square root of both sides of the equation. Remember that taking the square root will yield both positive and negative results:
[tex]\[ x = \pm\sqrt{16} \][/tex]
### Step 4: Simplify the square root
Simplifying the square root of 16, we get:
[tex]\[ x = \pm 4 \][/tex]
Thus, the solutions to the equation [tex]\(3x^2 - 48 = 0\)[/tex] are:
[tex]\[
x = 4 \quad \text{and} \quad x = -4
\][/tex]
### Conclusion
Considering the options given:
- a. [tex]\(\pm 4i\)[/tex]
- b. 4
- c. [tex]\(\pm 3 \sqrt{5}\)[/tex]
- d. [tex]\(3 \sqrt{5}\)[/tex]
None of these options are quite right on their own as written. However, the correct simplified solution to our equation is indeed [tex]\(4\)[/tex] and [tex]\(-4\)[/tex], which aligns with option b when combined as [tex]\(\pm 4\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{\pm 4} \][/tex]
