GMATinsight wrote:
What is the sum of all the real values of x for which |x-4|² + |x-4| = 30?
A) 16
B) 11
C) 9
D) 8
E) 7
Source:
http://www.GMATinsight.comTo make things easier, we'll use a technique known as
u-substitutionLet
u = |x - 4|So,
|x-4|² +
|x-4| = 30...
...becomes:
u² +
u = 30
Set this quadratic equal to zero: u² + u - 30 = 0
Factor: (u + 6)(u - 5) = 0
So, either u = -6 or u = 5
Since
u = |x - 4|, we can now that that either |x - 4| = -6 or |x - 4| = 5
Let's examine each case.
|x - 4| = -6
Since the absolute value is always greater than or equal to 0, it's IMPOSSIBLE for |something| = -6
So, this equation has no solution
|x - 4| = 5
This means that x - 4 = 5 or x - 4 = -5
If x - 4 = 5, then
x = 9If x - 4 = -5, then
x = -1So, the SUM OF THE SOLUTIONS =
9 + (
-1) = 8
Answer:
Cheers,
Brent
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