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What is the product of all the solutions of x^2+4x+7=|x+2|+3?

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What is the product of all the solutions of x^2+4x+7=|x+2|+3? [#permalink]

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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

My solution looked like this:

If x<0, then x^2+4x+7=-(x+2)+3 --->x^2+4x+7=-x-2+3---> (x+2)(x+3)=0 -->x=-2;-3

If x>=0, then x^2+4x+7=(x+2)+3--->x^2+4x+7=x+2+3--->(x+2)(x+1)=0 -->x=-2,-1, neither of the solutions are valid because for this case x>=0.

So, for my answer I got 6, which was incorrect. Obviously, my mistake was here: "x=-2,-1 -->neither of the solutions are valid because for this case x>=0", could anybody explain to me why am I wrong?
[Reveal] Spoiler: OA

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Re: What is the product of all the solutions of x^2+4x+7=|x+2|+3? [#permalink]

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viktorija wrote:
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

My solution looked like this:

If x<0, then x^2+4x+7=-(x+2)+3 --->x^2+4x+7=-x-2+3---> (x+2)(x+3)=0 -->x=-2;-3

If x>=0, then x^2+4x+7=(x+2)+3--->x^2+4x+7=x+2+3--->(x+2)(x+1)=0 -->x=-2,-1, neither of the solutions are valid because for this case x>=0.

So, for my answer I got 6, which was incorrect. Obviously, my mistake was here: "x=-2,-1 -->neither of the solutions are valid because for this case x>=0", could anybody explain to me why am I wrong?



there will be two cases .
1. wen mod of (x+2)<0
x<-2
So (x+2)(x+3)=0; x= -3
2. Wen mod of (x+2)>=0
x>=-2
so (x+2)(x+1)=0; x can take both values -1 &-2
combining both cases product will be -1*-2*-3=-6

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Re: What is the product of all the solutions of x^2+4x+7=|x+2|+3? [#permalink]

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New post 22 Dec 2014, 00:02
x^2+4x+7=|x+2|+3

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

(X+2)^2 = |x+2|
Only 3 values possible,
x = -3, x = -2, x =-1
So multiplication = -6

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Re: What is the product of all the solutions of x^2+4x+7=|x+2|+3? [#permalink]

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New post 22 Dec 2014, 02:53
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viktorija wrote:

My solution looked like this:

If x<0, then x^2+4x+7=-(x+2)+3 --->x^2+4x+7=-x-2+3---> (x+2)(x+3)=0 -->x=-2;-3


for x<0, possible values of x are -1,-2,-3....... now. for x=-1, |x+2| will be positive. hence the range you have selected for this equation is wrong. it should be x<-2

Quote:
If x>=0, then x^2+4x+7=(x+2)+3--->x^2+4x+7=x+2+3--->(x+2)(x+1)=0 -->x=-2,-1, neither of the solutions are valid because for this case x>=0.


by taking x>0. you have neglected the case of x=-1 for which |x+2| will be positive. thus your desired range should be x>-2.

you can read more about the modulus functions here. http://gmatclub.com/forum/math-absolute-value-modulus-86462.html

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Re: What is the product of all the solutions of x^2+4x+7=|x+2|+3? [#permalink]

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New post 22 Dec 2014, 03:26
viktorija wrote:
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

My solution looked like this:

If x<0, then x^2+4x+7=-(x+2)+3 --->x^2+4x+7=-x-2+3---> (x+2)(x+3)=0 -->x=-2;-3

If x>=0, then x^2+4x+7=(x+2)+3--->x^2+4x+7=x+2+3--->(x+2)(x+1)=0 -->x=-2,-1, neither of the solutions are valid because for this case x>=0.

So, for my answer I got 6, which was incorrect. Obviously, my mistake was here: "x=-2,-1 -->neither of the solutions are valid because for this case x>=0", could anybody explain to me why am I wrong?


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Re: What is the product of all the solutions of x^2+4x+7=|x+2|+3? [#permalink]

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New post 22 Dec 2014, 05:12
Two cases for the RHS viz (x+5) and (-x+1).
Equating to LHS gives sols as -1, -2, -3
Product = -6

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Re: What is the product of all the solutions of x^2+4x+7=|x+2|+3? [#permalink]

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New post 22 Dec 2014, 12:49
manpreetsingh86 wrote:
viktorija wrote:

My solution looked like this:

If x<0, then x^2+4x+7=-(x+2)+3 --->x^2+4x+7=-x-2+3---> (x+2)(x+3)=0 -->x=-2;-3


for x<0, possible values of x are -1,-2,-3....... now. for x=-1, |x+2| will be positive. hence the range you have selected for this equation is wrong. it should be x<-2

Quote:
If x>=0, then x^2+4x+7=(x+2)+3--->x^2+4x+7=x+2+3--->(x+2)(x+1)=0 -->x=-2,-1, neither of the solutions are valid because for this case x>=0.


by taking x>0. you have neglected the case of x=-1 for which |x+2| will be positive. thus your desired range should be x>-2.

you can read more about the modulus functions here. http://gmatclub.com/forum/math-absolute-value-modulus-86462.html


Thank you, manpreetsingh86, for the explanation! Now I see where my mistake was.

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Re: What is the product of all the solutions of x^2+4x+7=|x+2|+3? [#permalink]

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New post 22 Dec 2014, 14:51
Hi viktorija,

The "tricky" part of this question is the absolute value. When that type of math is involved, Test Takers often miss out on some of the possible answers, so it's important to do whatever you have to do to be sure that you've found all of the possibilities. To that end, I'm going to use ALL of the little clues that question gives me and avoid as much "crazy" math as possible.

We're given X^2+4X+7=|X+2|+3 and we asked for the product of ALL of the solutions.

Notice how the question asks us to find ALL of the solutions and NOT "both" solutions. This gets me thinking that there's probably more than 2 solutions. Also, quadratic equations usually have 1 or 2 solutions and absolute values usually have 1 or 2 solutions, so I'd be looking for up to 3 or 4 possibilities....

First, we simplify:

X^2+4X+7 = |X+2|+3
X^2+4X+4 = |X+2|
(X+2)(X+2) = |X+2|

Now, looking at that last line, there's NO WAY that x can be positive. Try TESTing any positive number - the two sides will NEVER equal. So there are NO positive answers.

Looking at the answer choices, they're all within a relatively small distance from 0, so the individual values for X are probably all relatively close to 0 as well. Let's find them with a bit of "brute force"...

X = 0
(0+2)(0+2) = |0+2|?
(2)(2) = 2? NO, 0 is NOT a solution

X = -1
(-1+2)(-1+2) = |-1+2|
(1)(1) = 1? YES, -1 IS a solution

X = -2
(-2+2)(-2+2) = |-2+2|
(0)(0) = 0? YES, -2 IS a solution

Remember that there were probably going to be MORE than 2 solutions, so we have to keep looking...

X = -3
(-3+2)(-3+2) = |-3+2|
(-1)(-1) = 1? YES, -3 IS a solution

At this point, if we take the product of these 3 answers we have (-1)(-2)(-3) = -6. Looking at the 5 answer choices, there CAN'T be any other answers. To go from -6 to -12, (2) would have to be an answer (but we already know that positive numbers DON'T FIT). To go from -6 to +12, (-2) would have to be used twice, which is not a mathematical option. So -6 MUST be the answer.

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Re: What is the product of all the solutions of x^2+4x+7=|x+2|+3? [#permalink]

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Re: What is the product of all the solutions of x^2+4x+7=|x+2|+3? [#permalink]

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Re: What is the product of all the solutions of x^2+4x+7=|x+2|+3? [#permalink]

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New post 14 Sep 2017, 05:46
I missed the (-1) root - silly mistake.
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Re: What is the product of all the solutions of x^2+4x+7=|x+2|+3? [#permalink]

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viktorija wrote:
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


Let’s first solve for when x + 2 is positive:

x^2 + 4x + 7 = x + 2 + 3

x^2 + 4x + 7 = x + 5

x^2 + 3x + 2 = 0

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

x = -2 or x = -1

Next we can solve for when x + 2 is negative:

x^2 + 4x + 7 = -(x + 2) + 3

x^2 + 4x + 7 = -x + 1

x^2 + 5x + 6 = 0

(x + 3)(x + 2) = 0

x = -3 or x = -2

We see that the product of all the solutions is (-2)(-1)(-3) = -6.

Answer: A
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Re: What is the product of all the solutions of x^2+4x+7=|x+2|+3?   [#permalink] 16 Sep 2017, 07:44
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