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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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GMATinsight
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What is the sum of all the real values of x for which |x-4|^2 + |x-4| [#permalink] New post   06 Jun 2017, 04:30
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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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BrentGMATPrepNow
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Re: What is the sum of all the real values of x for which |x-4|^2 + |x-4| [#permalink] New post   06 Jun 2017, 09:33
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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: https://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

Answer:
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D


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Re: What is the sum of all the real values of x for which |x-4|^2 + |x-4| [#permalink] New post   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] New post   06 Jun 2017, 06:14
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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: https://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] New post   06 Jun 2017, 09:25
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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: https://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)

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Re: What is the sum of all the real values of x for which |x-4|^2 + |x-4| [#permalink] New post   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: https://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.
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Re: What is the sum of all the real values of x for which |x-4|^2 + |x-4| [#permalink] New post   18 Oct 2021, 16:23
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: https://www.GMATinsight.com

But why can't the answer be C. Upon entering the value 9 in x, we get 25+5=30.

Then why not C? And what's wrong with my method? Please throw some light.

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Re: What is the sum of all the real values of x for which |x-4|^2 + |x-4| [#permalink] New post   19 Oct 2021, 00:33
The first thing I spot is if you move 30 over and set the expression = 0

[x - 4]^2 + [x - 4] - 30 = 0


+6 and -5 are two roots that multiply to -30 and add to +1


(1st) let the absolute value expression = A

[X - 4] = A everywhere the expression appears

(A)^2 + A - 30 = 0

(A + 6) (A - 5) = 0

A = -6 or A = +5


(2nd) now solve for the Modulus

Either

[X - 4] = -6

Or

[X - 4] = +5


Since the output of an absolute value expression can never be negative, the only valid root is when the Modulus expression = +5

[X - 4] = +5

Translation: “on the real number line, X is exactly 5 units away from +4”


X = -1 or +9

Sum: 9 - 1 =

8
Answer (D)

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Re: What is the sum of all the real values of x for which |x-4|^2 + |x-4| [#permalink] New post   19 Oct 2021, 02:27
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: https://www.GMATinsight.com


For questions on absolute values, you should always check the value obtained in the equation.
For example, after solving the above, we get 2 equations,

x^2 - 7x -18 = 0 and x^2 -9x -10 = 0
This gives x as -2, 9, -1 and 10
But if we plug these values back into the actual equation, 9 and -1 work while the others don't equate.
The real values are 9 and -1 and the sum is 8 D

Please give kudos if you like, stuck at 99

Thanks
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What is the sum of all the real values of x for which |x-4|^2 + |x-4| [#permalink] New post   25 Oct 2022, 03:00
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: https://www.GMATinsight.com



Ley |x-4| = y (where y is always positive) ......(a)

Equation becomes,
\(y^2 + y = 30\)
=> \(y^2 + y - 30 = 0\)
=> \(y=5\) (Ignoring the negative value as y cannot be negative) .....(b)

From (a) and (b),
\(|x-4| = 5\\
=> x-4 = +/- 5\\
=>x = -1,9\)

thus, the sum of all the real values of x for which \(|x-4|^2 + |x-4| = 30 is -1+9 = 8\)

D is Correct
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Re: What is the sum of all the real values of x for which |x-4|^2 + |x-4| [#permalink] New post   26 Oct 2022, 10:40
another way to solve this question:
|x-4|>0 must be true, let it equals to y
then x-4= +y and x-4=-y
therefore
x=+y+4 and x=-y+4
so add them up to get the sum of x values
+y+4-y+4= 8
so choose D.
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What is the sum of all the real values of x for which |x-4|^2 + |x-4| [#permalink] New post   19 Mar 2024, 02:33
­Here's a simpler approach:

Let |x-4| = a

Then, \(a^2 + a = 30\)
\(a(a+1) = 30\)

What two consecutive numbers when multiplied together will result in 30? 5*6 and -5*-6, correct!

Therefore, a = 5 or -5, and (x-4) = 5 or -5. Solve and we get x = 9 or -1. 9 + (-1) = 8

 ­
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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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