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Chapter 0: Problem 120

Solve each radical equation. Check all proposed solutions. $$\sqrt{6 x+1}=x-1$$

Short Answer

Expert verified
The solutions for the given radical equation are \(x=0\) and \(x=8\).

Step by step solution

01

Isolation of the Square Root

Start by isolating the square root term on one side of the equation. It is already isolated in this case: \(\sqrt{6x+1}=x-1\)
02

Square Both Sides

Next, to eliminate the square root, square both sides of the equation: \( (\sqrt{6x+1})^2 = (x-1)^2\). This becomes \(6x+1=x^2-2x+1\)
03

Solve Quadratic Equation

Rearrange the equation into standard quadratic form: \(x^2-2x-6x+1-1=0\). This simplifies to \(x^2-8x=0\). Factoring, we obtain \(x(x-8)=0\).
04

Find Values of x

Set each factor equal to zero to solve for x: \(x=0\) or \(x-8=0\), so \(x=8\). Thus, the two proposed solutions are \(x=0\) and \(x=8\)
05

Check Proposed Solutions

Plug the proposed solutions back into the original equation to ensure they do not violate the conditions of the square root. When plugged in, the original equation holds true for both established values.

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