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
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:
\[
\boxed{6}
\]
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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