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What is the sum of all roots of the equation [#permalink]
 29 Oct 2009, 02:30
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What is the sum of all roots of the equation
|x + 4|^2 - 10|x + 4| = 24?Please help me find my mistake: Let x+4=yNow we get two cases, Case1: y^2-10y-24=0Solving we get -2,12 Case2: -y^2+10y-24=0where we get 6,4
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Re: sum of all roots of the equation [#permalink]
29 Oct 2009, 03:11
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tejal777 wrote: What is the sum of all roots of the equation
|x + 4|^2 - 10|x + 4| = 24?
Please help me find my mistake:
Let x+4=y
Now we get two cases,
Case1:
y^2-10y-24=0
Solving we get -2,12
Case2:
-y^2+10y-24=0
where we get 6,4 This is good question. Let me show you how I've solved, maybe it'll help: We have |x + 4|^2 - 10|x + 4| = 24 |x + 4| flip sign at x=-4, so we should check two ranges: 1. x<=-4 (x+4)^2 + 10x+40=24 ((x+4)^2 as it's square will be the same in both ranges) x^2+8x+16+10x+16=0 --> x^2+18x+32=0. Solving for x: x=-16 or x=-2. x=-2 won't work as x<=-4 (see the defined range), hence we have only one solution for this range x=-16. 2. x>-4 (x+4)^2 - 10x-40=24 --> x^2-2x-48=0. Solving for x: x=-6 or x=8. x=-6 wont work as x>-4, hence we have only one root for this range x=8. -16+8=-8. Answer: the sum of all roots of the equation is -8.
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Re: sum of all roots of the equation [#permalink]
29 Oct 2009, 06:37
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Correct me:
I solved from where the author of the problem left it. that is:
y = -2 or 12
Hence, considereding + values of |x+4|, i.e. x+4 = -2 or 12, which gives us x = -6 or 8
Considering - values of |x+4|, i.e. -x-4 = -6 or 4, which gives us x = -2 or 8.
Sum of all, -6+8-2+8 = 8.
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Re: sum of all roots of the equation [#permalink]
29 Oct 2009, 13:32
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I solve it by replacing |x+4| as k
so k^2-10k-24=0
(k-12)(k+2) = 0
k = 12 or -2
k can not be -2 because it is an absolute value
so
k = 12 = |x+4|
then
x+4 = 12, x = 8
x+4 = -12, x = -16
sum is -8
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Re: sum of all roots of the equation [#permalink]
31 Oct 2009, 05:11
IMO -16.
Take y = |x+4 | and solve for y, then solve for |x+4| , we get x=-16,8,-2,-6, sum = -16.
OA?
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Re: sum of all roots of the equation [#permalink]
31 Oct 2009, 21:59
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Re: sum of all roots of the equation [#permalink]
07 Oct 2010, 12:19
mxgms wrote: jzd wrote: I solve it by replacing |x+4| as k
so k^2-10k-24=0
(k-12)(k+2) = 0
k = 12 or -2
k can not be -2 because it is an absolute value
so
k = 12 = |x+4|
then
x+4 = 12, x = 8
x+4 = -12, x = -16
sum is -8 I liked that approach, is this always true? thanks. Not sure what part you are questioning about (1) You can always do a variable switch in an equation (|x+4|=y) (2) |Any expression| is always greater than or equal to 0
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Re: sum of all roots of the equation [#permalink]
08 Oct 2010, 01:56
-8 for me. Once you solve the QE you get |x+4| = 6 or -4. -4 is not possible so take the case |x+4| = 6 which means x = -10 or 2. So the sum is -8.
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Re: sum of all roots of the equation [#permalink]
08 Dec 2010, 09:38
shrouded1 wrote: mxgms wrote: jzd wrote: I solve it by replacing |x+4| as k
so k^2-10k-24=0
(k-12)(k+2) = 0
k = 12 or -2
k can not be -2 because it is an absolute value
so
k = 12 = |x+4|
then
x+4 = 12, x = 8
x+4 = -12, x = -16
sum is -8 I liked that approach, is this always true? thanks. Not sure what part you are questioning about (1) You can always do a variable switch in an equation (|x+4|=y) (2) |Any expression| is always greater than or equal to 0 Shrouded: can we do this question by the approach you have mentioned in the walker post. .i.e |x-a|<b =>
a-b<x<a+b or this approach is for specific questions. Thanks
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Re: sum of all roots of the equation [#permalink]
14 Jun 2011, 02:24
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|x+4| = y gives y^2 -10y -24 = 0 y = -2 and 12 |x+4| = 12 gives x = 8 and -16. sum is -8.
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Re: What is the sum of all roots of the equation [#permalink]
17 Apr 2012, 07:04
Hi Buneuel,
Please help me with the basic understanding of the mod probs
when we have
|x+4|= |x-5|
We can two solutions
x+4= x-5
and
x+4=-x+5
but in this problem why do we check for ranges. I mean what s the step by step approach to attack a modulus question?
a few examples would be grateful
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Re: What is the sum of all roots of the equation [#permalink]
18 Apr 2012, 04:12
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Re: What is the sum of all roots of the equation [#permalink]
08 Sep 2012, 01:10
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tejal777 wrote: What is the sum of all roots of the equation
|x + 4|^2 - 10|x + 4| = 24?
Please help me find my mistake:
Let x+4=y
Now we get two cases,
Case1:
y^2-10y-24=0
Solving we get -2,12
Case2:
-y^2+10y-24=0
where we get 6,4 Let's try this with number line. |x+4| = y ==> y^2-10y-24=0 ==> y = 12 or y = -2 Substitute the value of y we have |x+4|=12 or |x+4|= -2 Hmm.. can mod be a negative number? NO ==> Eliminate |x+4|= -2 Now we are left only with |x+4|=12 Lets draw a number line .................................|x+4| ................................. <------------------------------------------------------------------------> -16 ..............................(-4) ................................8 Thus, two possible roots are -16 and +8 Sum of roots => -16+8=-8
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Re: What is the sum of all roots of the equation [#permalink]
08 Sep 2012, 03:20
tejal777 wrote: What is the sum of all roots of the equation
|x + 4|^2 - 10|x + 4| = 24?
Please help me find my mistake:
Let x+4=y
Now we get two cases,
Case1:
y^2-10y-24=0
Solving we get -2,12
Case2:
-y^2+10y-24=0
where we get 6,4 Case 1: You mean y\geq{0}, right? Because |x+4|=y only if y is non-negative. Only y=12 is acceptable. From |x+4|=12 we obtain x=8 and x=-16.Case 2: Now y<0, so |x+4|=-y. But |x+4|^2=(-y)^2 is still y^2, doesn't matter that y is negative! Your equation should be y^2+10y-24=0, solutions 2, -12. Now only -12 is acceptable ( y must be negative), and we obtain the same solutions as in Case 1. It would have been better to denote |x+4|=y\geq{0} (see other posts above). Then |x+4|^2=y^2, and for the quadratic equation y^2+10y-24=0 you choose only the non-negative root, then find x...
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Re: What is the sum of all roots of the equation [#permalink]
25 Sep 2012, 10:42
tejal777 wrote: What is the sum of all roots of the equation
|x + 4|^2 - 10|x + 4| = 24?
24 ends with 4 and 10|x+4| ends with 0. So |x+4|^2 should end with 4. Options below 0 are out because of the absolute value. Let's take the squares ending with 4: 2^2; 8^2; 12^2; 18^2 etc... We find 12^2 - 10*12 = 24. From here x_1=8 and x_2=-16, and the sum: -8.
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Re: What is the sum of all roots of the equation [#permalink]
06 Dec 2012, 03:12
It also took me time to understand how to get solutions for absolute values. But thanks to GMATClub... Here is a detailed explanation on how you could solve for the roots http://burnoutorbreathe.blogspot.com/2012/12/how-to-get-solution-for-absolute-values.htmlMy answer: -8
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Re: What is the sum of all roots of the equation [#permalink]
03 Feb 2013, 19:02
Have a question. There are two scenarios -
1. x<=-4
2. x>-4
How do we determine whether to include equal sign (=) in first equation or second equation or does it not matter and can be included in any?
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Re: What is the sum of all roots of the equation [#permalink]
04 Feb 2013, 04:12
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Re: What is the sum of all roots of the equation
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04 Feb 2013, 04:12
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