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# What is the sum of all the real values of x for which |x-4|^2 + |x-4|

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What is the sum of all the real values of x for which |x-4|^2 + |x-4|  [#permalink]

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06 Jun 2017, 04:30
17
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Difficulty:

85% (hard)

Question Stats:

42% (02:01) correct 58% (02:32) wrong based on 132 sessions

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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: www.GMATinsight.com

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Re: What is the sum of all the real values of x for which |x-4|^2 + |x-4|  [#permalink]

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06 Jun 2017, 09:33
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Top Contributor
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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.com

To make things easier, we'll use a technique known as u-substitution

Let 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 = 9
If x - 4 = -5, then x = -1

So, the SUM OF THE SOLUTIONS = 9 + (-1) = 8

Cheers,
Brent
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Re: What is the sum of all the real values of x for which |x-4|^2 + |x-4|  [#permalink]

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06 Jun 2017, 04:34
It should be D. As the two values which suffice are 9 and -1 after solving the quadratic.

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Re: What is the sum of all the real values of x for which |x-4|^2 + |x-4|  [#permalink]

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06 Jun 2017, 06:14
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.com

Checking options, i got Answer C. x = 9.

$$|x-4|^2$$ + |x-4| = 30
$$|9-4|^2$$ + |9 - 4| = $$5^2$$ + 5 = 25 + 5 = 30
Therefore x = 9. Answer C...
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Re: What is the sum of all the real values of x for which |x-4|^2 + |x-4|  [#permalink]

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06 Jun 2017, 09:25
2
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.com

Assume y=|x-4|
y^2+y-30=0
(y+6)(y-5)=0
y=-6 or y =5

For y=-6. |x-4|=-6
If x-4>=0 , x-4=-6, thus x=-2 (not ok)
If x-4<0 , x-4 = 6. thus x=10 (not ok)

For y=5. |x-4|=5
If x-4>=0 , x-4=5, thus x=9 (ok)
If x-4<0 , x-4 =-5. thus x=-1 (ok)

Sum of x = 9+(-1) = 8 (D)

Kudos if it helps
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Re: What is the sum of all the real values of x for which |x-4|^2 + |x-4|  [#permalink]

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15 May 2019, 05:17
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.com

Let |x-4| = t, so t>=0

Now t^2 + t -30=0

solving t= 5/-6
T can only assume non negative value, so...

|x-4|=5.
x can be -1 and 9.

Sum of all real values= 9-1=8.
Re: What is the sum of all the real values of x for which |x-4|^2 + |x-4|   [#permalink] 15 May 2019, 05:17
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