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Manager  G
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GMAT 1: 690 Q49 V35 If integers p and q are the roots of the equation ax^2 + bx + c = 0, w  [#permalink]

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Question Stats: 42% (02:32) correct 58% (02:32) wrong based on 224 sessions

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If integers p and q are the roots of the equation ax^2 + bx + c = 0, where a, b and c are constants and a > 0, by what percentage is c greater than |b|?

(1) |p+1| = |q – 3|
(2) The greatest number that divides both |p| and |q| is 2 and the smallest number that is divisible by both |p| and |q| is 12

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Re: If integers p and q are the roots of the equation ax^2 + bx + c = 0, w  [#permalink]

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Product of LCM and HCF of two numbers is equal to the product of the numbers

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Manager  B
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Re: If integers p and q are the roots of the equation ax^2 + bx + c = 0, w  [#permalink]

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@bunuel...Can u suggest the best approach, not able to understand this one

If integers p and q are the roots of the equation ax^2 + bx + c = 0, where a, b and c are constants and a > 0, by what percentage is c greater than |b|?

(1) |p+1| = |q – 3|
(2) The greatest number that divides both |p| and |q| is 2 and the smallest number that is divisible by both |p| and |q| is 12
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Re: If integers p and q are the roots of the equation ax^2 + bx + c = 0, w  [#permalink]

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There is a mistake in a question, it shouldn't be by what percent c is greater than |b|, I suppose it should be just a different or something like that, if so, the answer is C

Before moving to the statements I would like to mention that -p*-q=c and -p-q = b

Statement 1: Not sufficient. Can be any number. Ex: 12 and -19, |12+1|=|-10-3| or let's say -5 and -1, |-5+1|= |-1-3|

Statement 2: Again not sufficient. From this statement, we can find the value of |p|*|q| which is GCF*LCM= 2*12=24, from here we can have the following combinations of values of P and Q: (12, 2), (-12 , 2), (-12,- 2), (-2 , 12), (4, 6), (-4 ,-6), (-4, 6) and (-6, 4)

Combining this 2 statements together we can say that P and Q must be equal to (-4, 6) as this is the only pair that satisfies the statement 1. |6+1|=|-4-3| 7=7. Now we can calculate the C and b: C=-6*4=-24 b=-6+4=-2 and |b|=2
C is not greater than |b|. However, you cannot find the percent difference between these 2 numbers as one of them is negative, while the other is positive
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Re: If integers p and q are the roots of the equation ax^2 + bx + c = 0, w  [#permalink]

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Can anyone explain this problem a bit more clearly
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If integers p and q are the roots of the equation ax^2 + bx + c = 0, w  [#permalink]

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We have quadratic equation ax^2 + bx + c = 0(where a, b and c are constants and a>0)
We need to find out what percentage is 'c' greater than |b|(absolute value of b). Also known are the roots of the roots are p and q.

(1) |p+1| = |q–3|
Assume values for p and q which satisfy the equation.
Example 1 : p=+2,q=+5. A quadratic equation can be formed by (x-p)(x-q)
(x-2)(x-5) => x^2 -7x +10 = 0. Here c(10) is around 42.85% greater than |b|, which is 7.
Example 2 : p=+2,q=0. A quadratic equation can be formed by (x-p)(x-q)
(x-2)(x) => x^2 -2x +1 = 0 Here c(1) is around 50% lesser than |b|, which is 2. Clearly insufficient.

(2) The greatest number that divides both |p| and |q| is 2 and the smallest number that is divisible by both |p| and |q| is 12
Example 1 : p=-2,q=-6 A quadratic equation can be formed by (x-p)(x-q)
(x+2)(x+6) => x^2+4x+12 Here c(12) is around 200% greater than |b|, which is 4.
Example 2: p=-4,q=-6 A quadratic equation can be formed by (x-p)(x-q)
(x+4)(x+6) => x^2+10x+24 Here c(24) is around 140% greater than |b|, which is 10.Clearly insufficient.

When we combine both the statements, The only value possible for p & q, is p=-4,q=6

When p=-4,q=6 the conditions in both the statements are satisfied and the quadratic equation x^2 - 2x - 24=0
This has value of c(24) which is 1100% greater than |b|. Hence the combination of these statements is sufficient(Option C)
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If integers p and q are the roots of the equation ax^2 + bx + c = 0, w  [#permalink]

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We know that, Sum of the roots= p+q= -b/a AND Product of the roots= pq= c/a

From statement 1, |p+1|= |q-3| , i have taken 2 cases;
Case1- let p=4, q=8; then -b/a= 12, c/a=32
Case2- let p=-5 , q= 7; then -b/a= 2, c/a= -35
not sufficient as we get different answers

From statement 2, HCF of |p| and |q| =2 and LCM of |p| and |q| =12
Different combinations are possible= (12, 2), (-12,- 2), (-12 , 2), (-2 , 12), (4, 6), (-4 ,-6), (-4, 6) and (-6, 4)
not sufficient as we get different answers

Combining both statements,
out of the different combinations in statement 2, only two values satisfy
case1- p=6, q=-4 ; |p+1|= |q-3|; |6+1|= |-4-3|; 7=7
so p+q= 6-4=2= -b/a and pq= -6 * 4= -24= c/a

case2- p=-4, q=6 ; |p+1|= |q-3|; |-4+1|= |6-3|; 3=3
so p+q= -4+6=2= -b/a and pq= -4 * 6= -24= c/a

from both the cases, we can deduce that; -b=2a; |-b|= 2a and c=-24a
as a>0, so |b|= positive but c is negative
ideally, we will not be able to find the answer as c will not be greater than b
but according to me, we can write
(c-b) * 100/b
(-24a+2a) * 100/ 2a= -1100% i.e. c is greater than b by -1100%.

Please let me know if i am wrong?
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Re: If integers p and q are the roots of the equation ax^2 + bx + c = 0, w  [#permalink]

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GMATinsight Can you share an easier explanation to this question?
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Re: If integers p and q are the roots of the equation ax^2 + bx + c = 0, w  [#permalink]

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s111 wrote:
GMATinsight Can you share an easier explanation to this question?

s111 Here is my way of solving this question

Quote:
If integers p and q are the roots of the equation ax^2 + bx + c = 0, where a, b and c are constants and a > 0, by what percentage is c greater than |b|?

(1) |p+1| = |q – 3|
(2) The greatest number that divides both |p| and |q| is 2 and the smallest number that is divisible by both |p| and |q| is 12

Question REPHRASED: We need to know the sign and value of c and the absolute value of b to answer the question

Statement 1: |p+1| = |q – 3|

multiple solutions are possible hence
NOT SUFFICIENT

Statement 2: The greatest number that divides both |p| and |q| is 2 and the smallest number that is divisible by both |p| and |q| is 12
i.e. HCF of |p| and |q| = 2
and LCM of |p| and |q| = 12
Property: Product of two numbers = Product of their LCM and HCF
i.e. |p| * |q| = 2*12 = 24

Product of factors of any quadratic equation ax^2 + bx + c = 0 is given by c/a
i.e. product of roots here = +24 or -24 (as signs of p and q are still unknown)
still no information about b hence
NOT SUFFICIENT

Combining the two statements

We have |p+1| = |q – 3| and |p| * |q| = 24

24 may be written in following ways (i.e. values of |p| and |q| may be as follows in any order)
1*24 but this pair of values does NOT satisfy |p+1| = |q – 3|
2*12 but this pair of values do NOT satisfy |p+1| = |q – 3|
3*8 but this pair of values do NOT satisfy |p+1| = |q – 3|
4*6 this satisfies if p = 6 and q = -4

i.e. roots of the equation are 6 and -4
i.e. $$ax^2 + bx + c = (x-6)*(x+4)$$
i.e. $$ax^2 + bx + c = x^2 - 2x - 24$$

i.e. a = 1, b = -2 and c = -24
now we can calculate with CERTAINTY the percentage required hence
SUFFICIENT

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