To solve the equation [tex]\(x^2 = 12\)[/tex], follow these steps:
1. Understand the equation: We are given that [tex]\(x^2 = 12\)[/tex]. This means we need to find the value of [tex]\(x\)[/tex] that, when squared, equals 12.
2. Take the square root of both sides: To isolate [tex]\(x\)[/tex], take the square root on both sides of the equation. Remember, taking the square root of both sides will yield both positive and negative roots. So we have:
[tex]\[
x = \pm \sqrt{12}
\][/tex]
3. Simplify the square root: The number 12 can be factored into 4 and 3 (since 12 = 4 * 3), and since [tex]\(\sqrt{4} = 2\)[/tex], we can simplify:
[tex]\[
\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2 \sqrt{3}
\][/tex]
4. Include the positive and negative roots: This gives us two solutions:
[tex]\[
x = \pm 2\sqrt{3}
\][/tex]
Now we will compare these solutions to the given options:
- Option 1: [tex]\(x = \pm 4 \sqrt{3}\)[/tex]
- Option 2: [tex]\(x = \pm \sqrt{3}\)[/tex]
- Option 3: [tex]\(x = \pm 2 \sqrt{3}\)[/tex]
- Option 4: [tex]\(x = \pm 6 \sqrt{2}\)[/tex]
After comparing our simplified solution [tex]\(\pm 2 \sqrt{3}\)[/tex] with the given options, we find that option 3 ([tex]\(\pm 2 \sqrt{3}\)[/tex]) is the correct one.
Therefore, the correct answer is:
[tex]\[
x = \pm 2 \sqrt{3}
\][/tex]
This matches option 3 from the given choices.
