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Chapter 2: Problem 154

Solve: \(\sqrt{x+10}-4=x\)

Short Answer

Expert verified
The solution to the equation \(\sqrt{x+10}-4=x\) is \(x=-6\).

Step by step solution

01

Rearrange the equation

The goal for the first step is to rearrange the equation, isolating the square root. This leads to \(\sqrt{x+10}=x+4\)
02

Square both sides of the equation

To remove the square root, square both sides of the equation, resulting in \(x+10=(x+4)^2\). Now, perform the squaring operation, which gives \(x+10=x^2+8x+16\).
03

Rewrite equation to standard quadratic form

Rearrange the equation to the standard quadratic form \(ax^2+bx+c=0\). This results in \(x^2+7x+6=0\).
04

Factor the equation

The next step is to find factors of \(x^2+7x+6=0\). This leads to \((x+1)(x+6)=0\).
05

Solve for x

Setting each factor equal to zero and solving for x results in two solutions: \(x=-1\) and \(x=-6\).
06

Check for extraneous solutions

Since squaring in Step 2 might introduce extraneous solutions, it's vital to check if whether \(x=-1\) and \(x=-6\) are real solutions by replacing them in the original equation. We find that \(x=-1\) is not a solution but \(x=-6\) is, hence \(x=-6\) is the only solution to this equation.

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