Questions: Solve the radical equation. Check for extraneous solutions sqrt(x+5)=8

Solve the radical equation. Check for extraneous solutions sqrt(x+5)=8
Transcript text: Solve the radical equation. Check for extraneous solutions $\sqrt{x+5}=8$
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Solution

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Solution Steps

To solve the radical equation \(\sqrt{x+5} = 8\), we need to eliminate the square root by squaring both sides of the equation. This will give us a quadratic equation that we can solve for \(x\). After finding the potential solutions, we must check each one to ensure it is not extraneous by substituting back into the original equation.

Step 1: Eliminate the Square Root

To solve the equation \( \sqrt{x + 5} = 8 \), we square both sides to eliminate the square root: \[ x + 5 = 8^2 \] This simplifies to: \[ x + 5 = 64 \]

Step 2: Solve for \( x \)

Next, we isolate \( x \) by subtracting 5 from both sides: \[ x = 64 - 5 \] This gives us: \[ x = 59 \]

Step 3: Check for Extraneous Solutions

We substitute \( x = 59 \) back into the original equation to verify: \[ \sqrt{59 + 5} = \sqrt{64} = 8 \] Since this holds true, \( x = 59 \) is a valid solution.

Final Answer

The solution to the equation is \\(\boxed{x = 59}\\).

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