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Chapter 8: Problem 38

Solve each radical equation. $$x=\sqrt{3 x+7}-3$$

Short Answer

Expert verified
The solution to the equation \(x = \sqrt{3x + 7} - 3\) is \(x = -2\).

Step by step solution

01

Isolate the square root

Let's first isolate the square root to one side of the equation. We begin with the equation \(x = \sqrt{3x + 7} - 3\). It can be rearranged to \(\sqrt{3x + 7} = x + 3\)
02

Square both sides

Square both sides of equation to eliminate the square root. The equation becomes: \((x+3)^2 = (3x + 7)\) or \(x^2 + 6x + 9 = 3x + 7\)
03

Collecting similar terms

Move all terms from right-hand side to left-hand side to obtain, \(x^2 + 3x + 2 = 0\)
04

Factor

Factorize the left-hand side, \((x+2) \cdot (x+1) = 0\)
05

Find the roots

Set each factor equal to zero to get the value of \(x\). Which gives \(x = -2\) and \(x = -1\)
06

Checking for extraneous solutions

After solving the equation, we need to check for any extraneous solutions. The easiest way to do this is to substitute these solutions back into the original equation to see if they satisfy the original equation. For \(x = -1\), we get, \(-1 \neq \sqrt{3 \cdot (-1) + 7} - 3\). So \(x = -1\) is an extraneous solution and not a valid solution. For \(x = -2\), we get, \(-2 = \sqrt{3 \cdot (-2) + 7} - 3\). So \(x = -2\) is a valid solution.

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