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What is the product of all the solutions of (x + 2)^2 = |x + 2|?

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Bunuel
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What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   07 Aug 2023, 09:00
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What is the product of all the solutions of (x + 2)^2 = |x + 2|?

A. -6
B. -2
C. 2
D. 6
E. 12
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A
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Re: What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   07 Aug 2023, 09:22
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Bunuel wrote:
What is the product of all the solutions of (x + 2)^2 = |x + 2|?

A. -6
B. -2
C. 2
D. 6
E. 12


\(y = x + 2\)

\((y)^2 = |y|\)

squaring both sides

\(y^4 - y^2 = 0\)

\(y^2( y^2 - 1) = 0\)

  • Case 1: \(y^2 = 0\)
    • \(y = 0\)
      \(x + 2 = 0\)
      \(x = -2\)
  • Case 2 : \(y^2 - 1 = 0\)
    • \(y = +1\)
      \(x + 2 = +1\)
      \(x = -1\)
    • \(y = -1\)
      \(x + 2 = -1\)
      \(x = -3\)

Net product = \(-2 * -3 * -1 = -6\)

Option A
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Re: What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   27 Aug 2023, 06:45
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Bunuel wrote:
What is the product of all the solutions of (x + 2)^2 = |x + 2|?

A. -6
B. -2
C. 2
D. 6
E. 12


(x + 2)^2 = |x + 2|

|x + 2|^2 = |x + 2|

If the square of an expression is equal to the expression itself, then that expression must be equal to 1 or 0.

If |x + 2| = 1, then x = -1 OR x = -3.

If |x + 2| = 0, then x = -2.

So, the product of all the solutions is:

(-1)(-3)(-2) = -6

Answer: A

Note: In the solution, we used the rule that |y|^2 = y^2 for all y.
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Re: What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   07 Aug 2023, 10:42
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(x + 2)^2 = |x + 2|

case1: when x>=-2
(x + 2)^2 = (x + 2)
(x+2)*(x+2-1)=0
(x+2)*(x+1)=0
x=-2, -1

case2: when x<-2
(x + 2)^2 + (x + 2) =0
(x+2)*(x+3)=0
x=-3, x=-2
x cannot be equal to -2, since x<-2, hence x=-3 is the only root from case2

So roots are x=-1,-2 and -3
Product = -6, hence A should be the right choice.
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Re: What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   07 Aug 2023, 11:53
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visualizing plot of \(|x|=x^2\), solutions are \(x=-1,0,1\)

The function given in question is shifted left by \(2\) units, The new solutions will be \(x=-1-2,0-2,1-2=-3,-2,-1\)

Product =\(-6 \)
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Re: What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   08 Aug 2023, 01:09
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(x + 2)^2 = |x + 2|

x^2+4x+4 = lx+2l
we get values of x = -1,-2,-3
product is -6
option A

Bunuel wrote:
What is the product of all the solutions of (x + 2)^2 = |x + 2|?

A. -6
B. -2
C. 2
D. 6
E. 12
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Re: What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   07 Jan 2024, 14:54
Why can't we cancel (x+2)^4 = (x+2)^2 to make (x+2)^2 = 1?
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What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   07 Jan 2024, 23:40
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madboy1105 wrote:
Why can't we cancel (x+2)^4 = (x+2)^2 to make (x+2)^2 = 1?


madboy1105 - We can't divide on both sides of the equation by \((x+2)^2\) and arrive at \((x +2)^2 = 1\) as the value of \((x+2)^2\) can be zero (at x = -2).

Note: Whenever one cancels out value by dividing both sides by a common term, \((x+2)^2\) in this example, one needs to ensure that the divisor is not zero. Division by zero is an invalid operation.

As we cannot rule out that possibility in this case, the division operation is not valid and we cannot cancel out terms as you're expecting.

Hope this helps.
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What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   08 Jan 2024, 00:25
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(x + 2 )^2 = /x + 2 /

x^2 + 4x + 4 = /x + 2 /

Case 1

When x + 2 is positive.

x^2 + 4x + 4 = x + 2

x^2 + 4x + 4 -x - 2 = 0

x^2 + 3x + 2 = 0

x = -2
x = -1


Case 2


When x + 2 is negative.

x^2 + 4x + 4 = -x - 2

x^2 + 5x + 6 = 0

x = -2

x = -3

Substituting the values shows x could be -1, -2, -3

Sum = -6

Answer choice A
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Re: What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   18 Feb 2024, 10:20
adewale223 wrote:
(x + 2 )^2 = /x + 2 /

x^2 + 4x + 4 = /x + 2 /

Case 1

When x + 2 is positive.

x^2 + 4x + 4 = x + 2

x^2 + 4x + 4 -x - 2 = 0

x^2 + 3x + 2 = 0

x = -2
x = -1


Case 2


When x + 2 is negative.

x^2 + 4x + 4 = -x - 2

x^2 + 5x + 6 = 0

x = -2

x = -3

Substituting the values shows x could be -1, -2, -3

Sum = -6

Answer choice A

­
Shouldn´t case 2 factor to (x+1) (x+6)?? Hence x= -1 and x= -6. Their product is 6 and their difference is 5
 
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Re: What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   18 Feb 2024, 10:31
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ruis wrote:
adewale223 wrote:
(x + 2 )^2 = /x + 2 /

x^2 + 4x + 4 = /x + 2 /

Case 1

When x + 2 is positive.

x^2 + 4x + 4 = x + 2

x^2 + 4x + 4 -x - 2 = 0

x^2 + 3x + 2 = 0

x = -2
x = -1


Case 2


When x + 2 is negative.

x^2 + 4x + 4 = -x - 2

x^2 + 5x + 6 = 0

x = -2

x = -3

Substituting the values shows x could be -1, -2, -3

Sum = -6

Answer choice A

­
Shouldn´t case 2 factor to (x+1) (x+6)?? Hence x= -1 and x= -6. Their product is 6 and their difference is 5

 

­
When you expand (x + 1)(x + 6) do you get x^2 + 5x + 6? No. You get x^2 + 7x + 6.

Factoring Quadratics: https://www.purplemath.com/modules/factquad.htm
Solving Quadratic Equations: https://www.purplemath.com/modules/solvquad.htm

Hope it helps.­
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Re: What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   11 Mar 2024, 01:15
Hi Bunuel, I followed this method and ended up with the same results but did -2*-1*-3*-2. 

Question did not state distinct answers so 12 could be the answer as well.
adewale223 wrote:
(x + 2 )^2 = /x + 2 /

x^2 + 4x + 4 = /x + 2 /

Case 1

When x + 2 is positive.

x^2 + 4x + 4 = x + 2

x^2 + 4x + 4 -x - 2 = 0

x^2 + 3x + 2 = 0

x = -2
x = -1


Case 2


When x + 2 is negative.

x^2 + 4x + 4 = -x - 2

x^2 + 5x + 6 = 0

x = -2

x = -3

Substituting the values shows x could be -1, -2, -3

Sum = -6

Answer choice A

­
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Re: What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   11 Mar 2024, 02:33
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unicornilove wrote:
Hi Bunuel, I followed this method and ended up with the same results but did -2*-1*-3*-2. 

Question did not state distinct answers so 12 could be the answer as well.
adewale223 wrote:
(x + 2 )^2 = /x + 2 /

x^2 + 4x + 4 = /x + 2 /

Case 1

When x + 2 is positive.

x^2 + 4x + 4 = x + 2

x^2 + 4x + 4 -x - 2 = 0

x^2 + 3x + 2 = 0

x = -2
x = -1


Case 2


When x + 2 is negative.

x^2 + 4x + 4 = -x - 2

x^2 + 5x + 6 = 0

x = -2

x = -3

Substituting the values shows x could be -1, -2, -3

Sum = -6

Answer choice A

­

­
Firstly, the solutions of the equation are -1, -2, and -3. No one would say that the solutions are -1, -3, -2, AND -2 AGAIN.

Also, the method above lacks details to be 100% correct.

In case 1, when x + 2 is positive, it means the range is x > -2. So, you should discard x = -2, which you get for that case because it is not in the range.

In case 2, we should consider the scenario when x + 2 ≤ 0, giving the range x ≤ -2. Both x = -2 and x = -3 are in the range, so both are acceptable.

Thus, if done properly, that method gives only three roots: -1, -2, and -3.

Hope it helps.
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Re: What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   19 Mar 2024, 03:40
A algebraic way of calculating cases:

|x + 2| is the equivalent of ((x+2)^2)^1/2, so basically the square root of a squared number is equal to the absolute value

(x+2)^2 = ((x+2)^2)^1/2 => square
(x+2)^4 = (x+2)^2 => subtract - (x+2)^2
(x+2)^4 - (x+2)^2 =0 => Factorize (x+2)^2 (You only need to factorize and not solve for difference in squares which you could also use here if you look closely cause you want the equation to be in a factorized form = 0 to use the property that one factor has to be 0 then
(x+2)^2 * (((x+2)^2-1)) = 0; Therefore (x+2)^2 = 0 or (x+2)^2-1 = 0
(x+2)^2 = 0 if x = -2
(x+2)^2-1 = 0 if x = -3 or x = -1
Solution = -1*-3*-2 = -6
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Re: What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   18 Apr 2024, 00:26
All means to include All solutions. So why are we not including -2 again?

For example, if a question asks to find sum of all solutions and roots are 1,2,3 and 3 do we say answer as 6 and not 9?

Because that logic applies for distinct roots. the ambiguity of "all" solutions is unclear here. If the question stated distinct, then the answer to the question would be -1, -2 and -3 but it says all. In many TTP questions, they have solved questions containing all roots as containing all repeated roots as well JeffTargetTestPrep
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Re: What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   18 Apr 2024, 00:48
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CapnSal wrote:
All means to include All solutions. So why are we not including -2 again?

For example, if a question asks to find sum of all solutions and roots are 1,2,3 and 3 do we say answer as 6 and not 9?

Because that logic applies for distinct roots. the ambiguity of "all" solutions is unclear here. If the question stated distinct, then the answer to the question would be -1, -2 and -3 but it says all. In many TTP questions, they have solved questions containing all roots as containing all repeated roots as well JeffTargetTestPrep

­You doubt is addressed here: 

https://gmatclub.com/forum/what-is-the- ... l#p3365689

The solutions of the equation are -1, -2, and -3. No one would say that the solutions are -1, -3, -2, AND -2 AGAIN. Also, note that this is sn official question and the wording is as precise as it gets .
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Re: What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   18 Apr 2024, 01:48
Bunuel wrote:
CapnSal wrote:
All means to include All solutions. So why are we not including -2 again?

For example, if a question asks to find sum of all solutions and roots are 1,2,3 and 3 do we say answer as 6 and not 9?

Because that logic applies for distinct roots. the ambiguity of "all" solutions is unclear here. If the question stated distinct, then the answer to the question would be -1, -2 and -3 but it says all. In many TTP questions, they have solved questions containing all roots as containing all repeated roots as well JeffTargetTestPrep

­You doubt is addressed here: 

https://gmatclub.com/forum/what-is-the- ... l#p3365689

The solutions of the equation are -1, -2, and -3. No one would say that the solutions are -1, -3, -2, AND -2 AGAIN. Also, note that this is sn official question and the wording is as precise as it gets .

­Thanks Bunuel, does this mean the "distinct" implication is implied ?­
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Re: What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   18 Apr 2024, 02:00
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CapnSal wrote:
­Thanks Bunuel, does this mean the "distinct" implication is implied ?­

­
The point is, if you solve the question correctly as demonstrated in many of the above posts, you don't even get x = -2 twice. Therefore, it doesn't even need to be implied.
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Re: What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink] New post   18 Apr 2024, 04:26
Asked: What is the product of all the solutions of (x + 2)^2 = |x + 2|?

(x+2)ˆ2 = |x+2|
|x+2|ˆ2 - |x+2| = 0
|x+2| ( |x+2| - 1) = 0
Case 1: |x+2| = 0
x = -2
Case 2: |x+2| - 1 =0
|x+2| = 1
x = -1 or -3

x = {-3,-2,-1}

The product of all the solutions = (-3)(-2)(-1) = -6

IMO A
­
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Re: What is the product of all the solutions of (x + 2)^2 = |x + 2|? [#permalink]
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