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# What is the sum of all possible solutions to |x - 3|^2 + |x - 3| = 20

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Manager
Joined: 16 Mar 2016
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GMAT 1: 660 Q47 V33
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What is the sum of all possible solutions to |x - 3|^2 + |x - 3| = 20  [#permalink]

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26 May 2016, 12:12
2
24
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Difficulty:

65% (hard)

Question Stats:

55% (02:03) correct 45% (02:30) wrong based on 324 sessions

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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
Math Expert
Joined: 02 Sep 2009
Posts: 61470
Re: What is the sum of all possible solutions to |x - 3|^2 + |x - 3| = 20  [#permalink]

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26 May 2016, 12:28
4
7
Alex75PAris 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

First of all |x - 3|^2 = (x - 3)^2, so we have: (x - 3)^2 + |x - 3| = 20.

When x < 3, x - 3 is negative, thus |x - 3| = -(x - 3). In this case we'll have (x - 3)^2 - (x - 3) = 20 --> x = -1 or x = 8. Discard x = 8 because it's not in the range we consider (< 3).

When x >= 3, x - 3 is non-negative, thus |x - 3| = x - 3. In this case we'll have (x - 3)^2 + (x - 3) = 20 --> x = -2 or x = 7. Discard x = -2 because it's not in the range we consider (>= 3).

Thus there are two solutions: x = -1 and x = 7 --> the sum = 6.

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Math Expert
Joined: 02 Sep 2009
Posts: 61470
Re: What is the sum of all possible solutions t  [#permalink]

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26 May 2016, 12:31
7
2
Alex75PAris 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

Another way.

Denote |x - 3| as y: y^2 + y = 20 --> y = -5 or y = 4. Discard the first solution since y = |x - 3|, so it's an absolute value and thus cannot be negative.

y = |x - 3| = 4 --> x = 7 or x = -1. The sum = 6.

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What is the sum of all possible solutions to |x - 3|^2 + |x - 3| = 20  [#permalink]

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24 Dec 2018, 14:15
Bunuel wrote:
Alex75PAris 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

Another way.

Denote |x - 3| as y: y^2 + y = 20 --> y = -5 or y = 4. Discard the first solution since y = |x - 3|, so it's an absolute value and thus cannot be negative.

y = |x - 3| = 4 --> x = 7 or x = -1. The sum = 6.

Hello Bunuel!

Why is it -5 and 4?

Is not as the following?

a2 + a - 20 = 0

(a + 5 ) (a - 4 ) = 0

Best regards!
Math Expert
Joined: 02 Sep 2009
Posts: 61470
Re: What is the sum of all possible solutions to |x - 3|^2 + |x - 3| = 20  [#permalink]

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24 Dec 2018, 19:15
jfranciscocuencag wrote:
Bunuel wrote:
Alex75PAris 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

Another way.

Denote |x - 3| as y: y^2 + y = 20 --> y = -5 or y = 4. Discard the first solution since y = |x - 3|, so it's an absolute value and thus cannot be negative.

y = |x - 3| = 4 --> x = 7 or x = -1. The sum = 6.

Hello Bunuel!

Why is it -5 and 4?

Is not as the following?

a2 + a - 20 = 0

(a + 5 ) (a - 4 ) = 0

Best regards!

(y + 5 )(y - 4 ) = 0

y + 5 = 0 --> y = -5.
y - 4 = 0 --> y = 4.

P.S. Why are you using 'a' if there is 'y' in the solution?
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GMAT 2: 640 Q49 V27
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Re: What is the sum of all possible solutions to |x - 3|^2 + |x - 3| = 20  [#permalink]

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24 Dec 2018, 20:51
2
1
$$|x - 3|^2 + |x - 3| = 20$$

Let |x-3| = a

$$a^2 + a - 20 = 0$$
$$a^2 + 5a - 4a - 20 = 0$$
$$a(a+5) - 4(a+5) = 0$$
$$(a-4)(a+5)=0$$
$$a = 4, -5$$

As |x-3| = a, |x-3| = 4, -5. But modulus of any value cannot be a negative integer. So |x-3| = 4

x-3 = 4, -4
x = 7, -1

Sum of possible solutions is => 7-1 => 6

OPTION: B
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Re: What is the sum of all possible solutions to |x - 3|^2 + |x - 3| = 20  [#permalink]

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08 Feb 2019, 11:40
Sorry i still don't understand the following

(x - 3)^2 - (x - 3) = 20 --> x = -1 or x = 8.

isn't it (x-3)(x-3)-(x-3)=20
(x^2)-6x+9-x+3=20
(x^2)-7x+12=20
(x^2)-7x=8
Bunuel - how did you factorize it to get -1 and 8. Was it just by plugging numbers in?
Math Expert
Joined: 02 Sep 2009
Posts: 61470
Re: What is the sum of all possible solutions to |x - 3|^2 + |x - 3| = 20  [#permalink]

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08 Feb 2019, 23:50
Albs wrote:
Sorry i still don't understand the following

(x - 3)^2 - (x - 3) = 20 --> x = -1 or x = 8.

isn't it (x-3)(x-3)-(x-3)=20
(x^2)-6x+9-x+3=20
(x^2)-7x+12=20
(x^2)-7x=8
Bunuel - how did you factorize it to get -1 and 8. Was it just by plugging numbers in?

$$(x - 3)^2 - (x - 3) = 20$$
$$x^2 - 6x + 9 - x + 3 = 20$$
$$x^2 - 7 x - 8 = 0$$

From here you can either factor to get (x - 4) (x - 3) = 20 or solve using discriminant.

7. Algebra

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.
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Re: What is the sum of all possible solutions to |x - 3|^2 + |x - 3| = 20   [#permalink] 08 Feb 2019, 23:50
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