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Chapter 0: Problem 118

Solve each radical equation. Check all proposed solutions. $$\sqrt{x+10}=x-2$$

Short Answer

Expert verified
The solution to the equation \( \sqrt{x+10}=x-2\) is \(x = 6\).

Step by step solution

01

Isolate the square root

In the given equation \( \sqrt{x+10}=x-2\), the square root is already isolated.
02

Square both sides of the equation

Squaring both sides, you get \(x+10 = (x-2)^2\). Expanding the square on the right-hand side, this simplifies to \(x+10 = x^2 - 4x + 4.\)
03

Rearrange the equation

Rearrange the equation to be equal to zero, \(x^2 - 5x - 6 = 0.\)
04

Factoring

Factor this equation. You obtain \((x-6)(x+1) = 0.\)
05

Find possible solutions

Setting each factor equal to zero gives two possible solutions, \(x=6\) and \(x=-1.\)
06

Check the solutions

Checking \(x=6\) in the original equation, you get \( \sqrt{6+10} = 6 - 2\) which simplifies to \(4 = 4\), a true statement. It means \(x=6\) is a valid solution. However, when checking \(x=-1\), you get \( \sqrt{-1+10}= -1 - 2\) which simplifies to \( \sqrt{9}= -3\), but \( \sqrt{9}\) equals to 3, not -3 so it is not a valid solution. Therefore, \(x=-1\) is an extraneous solution and is discarded.

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