What is the product of all the solutions of x^2 - 4x + 6=3

Question

What is the product of all the solutions of  x^2 - 4x + 6 = 3 - |x - 1|  ?

(A) -8
(B) -4
(C) 2
(D) 4
(E) 8

Edited the answer choice C from -2 to 2 and the OA. It's C not E.

Bunuel · Accepted Answer

If x<1 , then |x - 1| = -(x-1)=1-x , so in this case we'll have x^2 - 4x + 6 = 3-(1-x) --> x^2-5x+4=0 --> x=1 or x=4 --> discard both solutions since neither is in the range x<1 . If x\geq{1} , then |x - 1| = x-1 , so in this case we'll have x^2 - 4x + 6 = 3-(x-1) --> x^2-3x+2=0 --> x=1 or x=2 . Therefore, the product of the roots is 1*2=2. Answer: C.

fameatop · Answer

If |x - 1|>=0 ---->then the modulus will be equal to (x-1) & roots of the resulting equation will be 2,1
If |x - 1|<0 ---->then the modulus will be equal to (-x+1) & roots of the resulting equation will be 4,1

So the product of all the roots (2,1,4,1) is 8.

caioguima · Answer

In order for |x - 1| to be equal to 1 - x, we would have to have x < 1 . Therefore eliminating your second pair of solutions

You could also verify this by substituting x = 4 inthe original equation, and seeing that this solution DOES NOT fit. The only two solutions are 1 and 2.

ANS: no correct option available
   Posted from my mobile device

geometric · Answer

Wow! Really? How often do we see this? Who the heck wrote this question?

Bunuel · Answer

How often do we see what?

I've seen similar question which reads:
 What is the product of all the solutions of x^2 + 4x + 7 = |x + 2| + 3 ? 
A. -6 
B. -2 
C. 2 
D. 6 
E. 12

OA:  A

Try it. I'll provide solution for this question later, if necessary.

geometric · Answer

How often do we see no correct answer among the answer choices?

davidfrank · Answer

I have a question, I do understand that why have you taken the value 1 but I don't understand why have you taken x>=1. Why not simply x>1

Bunuel · Answer

x could be 1, thus when you consider the ranges you should include this value in either of the range, so we could consider x<1 and x>=1 OR x<=1 and x>1 (you cam include = sign in either of the ranges). Hope it's clear.

Bunuel · Answer

Never, if it's a proper GMAT question.

guerrero25 · Answer

I've seen similar question which reads:
 What is the product of all the solutions of x^2 + 4x + 7 = |x + 2| + 3 ? 
A. -6 
B. -2 
C. 2 
D. 6 
E. 12

OA:  A

Try it. I'll provide solution for this question later, if necessary.

if |x+2|>=0 then |x+2|= (x+2)

eqation becomes (x+2)(x+1)=0

x=-2,-1

if |x+2|<0 then |x+2|=-(x+2)

equation becomes (x+2) (x+3) =0
x=-2,-3 ( can't be -2 since x<-2)

product of the solution -

-2*-1*-3= -6 Ans

Bunuel · Answer

I guess you meant the following:

When  x\leq{-2} , then  |x+2|=-(x-2) .
When  x>{-2} , then  |x+2|=(x-2) .

Complete solution:

x^2 + 4x + 7 = |x + 2| + 3  -->  x^2 + 4x + 4 = |x + 2|  -->  (x+2)^2=|x+2|  -->  (x+2)^4=(x+2)^2  -->  (x+2)^2((x+2)^2-1)=0 :

x+2=0  -->  x=-2 ;
OR
 (x+2)^2-1=0  -->  (x+2)^2=1  -->  x=-1  or  x=-3 .

The product of the roots:  (-2)*(-1)*(-3)=-6 .

Answer: A.

Hope it's clear.

guerrero25 · Answer

thanks ! That was a Typo . I edited the post .

Bunuel · Answer

if |x+2|>=0 then |x+2|= (x+2)

eqation becomes (x+2)(x+1)=0

x=-2,-1

if |x+2|<0 then |x+2|=-(x+2)

equation becomes (x+2) (x+3) =0
x=-2,-3 ( can't be -2 since x<-2)

product of the solution -

-2*-1*-3= -6 Ans

Still not correct. Absolute value cannot be negative, so |x+2| is always more than or equal to zero.

jjack0310 · Answer

Bunuel, 
Can you solve this problem using the other method that you used in the previous problem?

I mean:

If 
x >= 0, |x + 2| = x + 2.
This would give the equation: x^2 + 4x + 7 = x + 5.
Roots are -2, and -1

what is the other scenario? 
What happens if x < 0?

How do we end up with the roots -3, and -1??
Thaanks

Bunuel · Answer

When  x\leq{-2} , then  |x+2|=-(x-2) . So, in this case we'll have  x^2 + 4x + 7 =-(x + 2) + 3  -->  x=-3  or  x=-2 . Both solutions are valid.

When  x>{-2} , then  |x+2|=(x-2) . So, in this case we'll have  x^2 + 4x + 7 =(x + 2) + 3  -->  x=-2  or  x=-1 . The first solution is not valid since it's out of the range we consider. The second one is OK.

So, there are 3 valid solutions:  x=-3 ,  x=-2  and  x=-1 .

Hope it's clear.

jjack0310 · Answer

Sorry. It is not clear.

Can you explain what was wrong with the way I was approaching the problem?

I mean other than the part that you marked red, what was I doing wrong? Do I have to solve theproblem using the solution that you mentioned?

If x > -2, how is |x + 2| = (x - 2)?
Is there an identity that I am missing? 
If I plug in, X = -1, |x + 2| = 1, but (x - 2) = -3

Why the discrepancy? What identity am I missing?

Bunuel · Answer

There was a typo:
When  x\leq{-2} , then  |x+2|=-(x+2) 
When  x>{-2} , then  |x+2|=(x+2) .

Absolute value properties:

When  x\leq {0}  then  |x|=-x , or more generally when  some \ expression\leq{0}  then  |some \ expression|={-(some \ expression)} . For example:  |-5|=5=-(-5) ;

When  x \geq {0}  then  |x|=x , or more generally when  some \ expression\geq{0}  then  |some \ expression|={some \ expression} . For example:  |5|=5 .

For our question, when x>-2 (when x+2>0), |x+2|=x+2.

Hope it's clear.

jjack0310 · Answer

Got it.

Thanks,

Final question, why are there two possibilities for when x = 0? Is that correct? or a typo?

Bunuel · Answer

No, that's not a typo: |0|=0=-0.

What is the product of all the solutions of x^2 - 4x + 6=3

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What is the product of all the solutions of x^2 - 4x + 6=3

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