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

Question

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

Bunuel · Accepted Answer

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. Answer: B.

Bunuel · Answer

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.

Answer: B.

jfranciscocuencag · Answer

Hello Bunuel!

Why is it -5 and 4?

Is not as the following?

a2 + a - 20 = 0

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

Best regards!

Bunuel · Answer

(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?

eswarchethu135 · Answer

|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

Albs · Answer

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?

Bunuel · Answer

(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.

Check the links below:

Factoring Quadratics 
 Solving Quadratic Equations

7. Algebra 
     Theory   
 Math : Algebra 101 
 Algebra: Tips and hints 
 FOIL on the GMAT: Simplifying and Expanding 
 Factoring Quadratics 
 Solving Quadratic Equations  
 Using Algebra vs. Logic on GMAT Quant Questions 
     Questions   
 Tough Algebra Set 
 Algebra Ability Quiz by e-GMAT 
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

Hope it helps.

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

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