To find the solutions to the equation [tex]\(x^2 = 12\)[/tex], let's go through the solution step-by-step:
1. Understanding the Equation:
The equation [tex]\(x^2 = 12\)[/tex] expresses that [tex]\(x\)[/tex] squared is equal to 12. We need to find the values of [tex]\(x\)[/tex] that satisfy this equation.
2. Solving for [tex]\(x\)[/tex]:
We can solve the equation by taking the square root of both sides. Remember that taking the square root of both sides will yield both the positive and negative roots.
[tex]\[
x = \pm \sqrt{12}
\][/tex]
3. Simplifying [tex]\(\sqrt{12}\)[/tex]:
Next, we simplify [tex]\(\sqrt{12}\)[/tex]. We do this by factoring 12 into its prime components:
[tex]\[
12 = 4 \times 3 \quad \text{and} \quad 4 = 2^2
\][/tex]
Therefore,
[tex]\[
\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}
\][/tex]
4. General Solution:
Thus, the solutions to the equation [tex]\(x^2 = 12\)[/tex] are:
[tex]\[
x = \pm 2\sqrt{3}
\][/tex]
5. Matching with the Options Given:
Now, we compare these solutions with the options provided:
A. [tex]\(x = \pm 2 \sqrt{6}\)[/tex]
B. [tex]\(x = \pm 6 \sqrt{2}\)[/tex]
C. [tex]\(x = \pm 3 \sqrt{2}\)[/tex]
D. [tex]\(x = \pm 2 \sqrt{3}\)[/tex]
From our work, we determined that the solutions are:
[tex]\[
x = \pm 2\sqrt{3}
\][/tex]
These match exactly with Option D:
[tex]\[
x = \pm 2 \sqrt{3}
\][/tex]
So, the correct answer is Option D: [tex]\(x = \pm 2\sqrt{3}\)[/tex].
