Question

Solve the square root equation 2x−−√=x+6−−−−−√.(1 point)
Responses

x=65
x equals Start Fraction 6 over 5 End Fraction

x=6
x equals 6

x=2
x equals 2

x=−6

Answers

Answered by GPT-4o mini
To solve the equation \(\sqrt{2x} = \sqrt{x + 6}\), we will first square both sides to eliminate the square roots.

1. Start with the original equation:
\[
\sqrt{2x} = \sqrt{x + 6}
\]

2. Square both sides:
\[
(\sqrt{2x})^2 = (\sqrt{x + 6})^2
\]

This simplifies to:
\[
2x = x + 6
\]

3. Now, rearrange the equation to isolate \(x\):
\[
2x - x = 6
\]
\[
x = 6
\]

Now we will check if \(x = 6\) is a valid solution by substituting it back into the original equation:

- Left-hand side:
\[
\sqrt{2(6)} = \sqrt{12} = 2\sqrt{3}
\]
- Right-hand side:
\[
\sqrt{6 + 6} = \sqrt{12} = 2\sqrt{3}
\]

Both sides are equal, so \(x = 6\) is indeed a solution.

The correct answer is:
\[
\boxed{6}
\]
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