skylimit wrote:
What is the sum of all possible solutions to \(|x-3|^2 - |x-3| = 20\)?
A) -1
B) 6
C) 7
D) 12
E) 14
Source: GMAT Prep Now -
http://www.gmatprepnow.com/module/gmat- ... video/1018 Important Property:\(\sqrt{x^2}=|x|\)
Since \(|x-3|=\sqrt{(x-3)^2}\), \(|x-3|^2= (x-3)^2\)
When \(x>3, |x-3|=x-3\) and when \(x<3, |x-3|=3-x\).
When \(x>3\), \((x-3)^2-(x-3)=20\) yields two solutions \(x=-1\) or \(x=8\), but since \(x=-1\) is not in the range \(x>3\), this solution should be rejected.
When \(x<3\), \((x-3)^2-(3-x)=20\) yields two solutions \(x=-2\) or \(x=7\), but since \(x=7\) is not in the range \(x<3\), this solution should be rejected.
The valid solutions are \(x=-2\) and \(x=8.\) Their sum is 6.