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Question of the week - 33 (A quadratic equation is in the form ......)

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Question of the week - 33 (A quadratic equation is in the form ......)  [#permalink]

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New post 25 Jan 2019, 05:22
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Question Stats:

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A quadratic equation is in the form of \(x^2 – 2px + m = 0\), where m is divisible by 5 and is less than 120. One of the roots of this equation is 7. If p is a prime number and one of the roots of the equation, \(x^2 – 2px + n = 0\) is 12, then what is the value of p + n – m?

    A. 0
    B. 6
    C. 16
    D. 26
    E. 27

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Re: Question of the week - 33 (A quadratic equation is in the form ......)  [#permalink]

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New post 25 Jan 2019, 06:53
Bunuel sir, This question has been posted twice.

https://gmatclub.com/forum/question-of-the-week-33-a-quadratic-equation-is-in-the-form-287275.html#p2214412

Can you merge the replies?
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Re: Question of the week - 33 (A quadratic equation is in the form ......)  [#permalink]

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New post 25 Jan 2019, 07:13
EgmatQuantExpert wrote:
A quadratic equation is in the form of \(x^2 – 2px + m = 0\), where m is divisible by 5 and is less than 120. One of the roots of this equation is 7. If p is a prime number and one of the roots of the equation, \(x^2 – 2px + n = 0\) is 12, then what is the value of p + n – m?

    A. 0
    B. 6
    C. 16
    D. 26
    E. 27



IMO D

prime numbers will be 2,3,5,11

Multiples of 5 which can give us a 7 => 35,70 and 105

\(x^2 – 2px + m = 0\), here p can be 11 and m will be 105

\(x^2 – 22x + m = 0\)
\(x^2 – 15x -7x + 105 = 0\)
\(x(x – 15) -7(x -15) = 0\)

\(x^2 – 2px + n = 0\), here p can be 11 and n will be 120
\(x^2 – 22x + 120 = 0\)
\(x^2 – 10x -12x +120 = 0\)
\(x(x – 10) -12(x - 10) = 0\)

so values of p + n – m
11 + 120 - 105
Answer D

P.S. I was down to realize that the value was 11(in 2 mins), and when i was writing this solution to ask some help on how to solve this question.

I didn't realize that the factors of 105 would have given me 22.

Always take all the possibilities of the factors
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Re: Question of the week - 33 (A quadratic equation is in the form ......)  [#permalink]

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New post 25 Jan 2019, 09:55
1
\(x^2 – 2px + m = 0\)

given one value of x=7
49-14p+m=0
p=2,3,5,7,11,13...
upon doing substitution for values of p only at 11 we get m as a integer which is divisible by 5
so
49-14*11+m=0
m= 105

\(x^2 – 2px + n = 0\)
at x=12
144-24p+n
p=11
n=120
so
p + n – m
11+120-105

26
IMO D


EgmatQuantExpert wrote:
A quadratic equation is in the form of \(x^2 – 2px + m = 0\), where m is divisible by 5 and is less than 120. One of the roots of this equation is 7. If p is a prime number and one of the roots of the equation, \(x^2 – 2px + n = 0\) is 12, then what is the value of p + n – m?

    A. 0
    B. 6
    C. 16
    D. 26
    E. 27

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Re: Question of the week - 33 (A quadratic equation is in the form ......)  [#permalink]

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New post 25 Jan 2019, 19:24
This is doable, but a very tricky question. Considering m could take 35, 70, and 105 was important, as the value of "m" was not 35, or 70, it was 105.
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Re: Question of the week - 33 (A quadratic equation is in the form ......)  [#permalink]

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New post 29 Jan 2019, 23:43

Solution


Given:
    • A quadratic equation, \(x^2 – 2px + m\) = 0
      o m is divisible by 5, and
      o m < 120
      o 7 is one root of the equation
      o p is a prime number
    • 12 is one root of the equation, \(x^2 – 2px + n = 0\)

To find:
    • The value of p + n – m

Approach and Working:
In the quadratic equation, \(x^2 – 2px + m = 0\), let us assume that the other root is “b”
    • Sum of the roots = 7 + b = 2p
      o Implies, 7 + b must be even, that is b must be odd

    • Product of the roots = 7b = m = a multiple of 5
      o Implies, b is a multiple of 5

    • Thus, b is an odd multiple of 5
    • And, we are given that m < 120
      o Implies, 7 * m = 7 * 5k < 120
      o \(k < \frac{24}{7}\)
      o Thus, k = 1 or 3

    • If k = 1, then b = 5
      o In this case, 2p = 7 + 5 = 12
      o p = 6, which is not prime

    • If k = 3, then b = 15
      o In this case, 2p = 7 + 15 = 22
      o p = 11, which is a prime number

    • So, p = 11, m = 7 * 15 = 105, and n = 12 * 10 = 120

Therefore, p + n – m = 11 + 120 – 105 = 26

Hence the correct answer is Option D.

Answer: D

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Re: Question of the week - 33 (A quadratic equation is in the form ......)  [#permalink]

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New post 31 Jan 2019, 07:27
Hello Archit3110 !

I was reading your approach and I liked it, I just have a question.

Why can we equal x to one of the roots? Would it be the same result if we equate x to the other unknown root?

I am a bit confused with that:

"given one value of x=7
49-14p+m=0"


Kind regards!
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Question of the week - 33 (A quadratic equation is in the form ......)  [#permalink]

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New post 31 Jan 2019, 07:31
EgmatQuantExpert wrote:

Solution


Given:
    • A quadratic equation, \(x^2 – 2px + m\) = 0
      o m is divisible by 5, and
      o m < 120
      o 7 is one root of the equation
      o p is a prime number
    • 12 is one root of the equation, \(x^2 – 2px + n = 0\)

To find:
    • The value of p + n – m

Approach and Working:
In the quadratic equation, \(x^2 – 2px + m = 0\), let us assume that the other root is “b”
    • Sum of the roots = 7 + b = 2p
      o Implies, 7 + b must be even, that is b must be odd

    • Product of the roots = 7b = m = a multiple of 5
      o Implies, b is a multiple of 5

    • Thus, b is an odd multiple of 5
    • And, we are given that m < 120
      o Implies, 7 * m = 7 * 5k < 120
      o \(k < \frac{24}{7}\)
      o Thus, k = 1 or 3

    • If k = 1, then b = 5
      o In this case, 2p = 7 + 5 = 12
      o p = 6, which is not prime

    • If k = 3, then b = 15
      o In this case, 2p = 7 + 15 = 22
      o p = 11, which is a prime number

    • So, p = 11, m = 7 * 15 = 105, and n = 12 * 10 = 120

Therefore, p + n – m = 11 + 120 – 105 = 26

Hence the correct answer is Option D.

Answer: D

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Hello EgmatQuantExpert

I was doing the same approach as you but I had a problem with the following:

If its given that one of the roots is 7, why it doesn't mean that x = -7 because if one root is 7, that would mean that:

(x -7) Isn't it? So the sum of the root -2px shouldn't be (-7) + b?

Could you please help me with this?

Thank you in advance!

Kind regards!
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Re: Question of the week - 33 (A quadratic equation is in the form ......)  [#permalink]

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New post 31 Jan 2019, 07:58
jfranciscocuencag wrote:
Hello Archit3110 !

I was reading your approach and I liked it, I just have a question.

Why can we equal x to one of the roots? Would it be the same result if we equate x to the other unknown root?

I am a bit confused with that:

"given one value of x=7
49-14p+m=0"


Kind regards!


jfranciscocuencag
its given in the question that one of the roots of equation is 7 .. so value of x has been substituted as 7.. by roots in quadratic eqn it means that upon doing substitution of the value of x we would get value "= 0"
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Re: Question of the week - 33 (A quadratic equation is in the form ......)  [#permalink]

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New post 01 Feb 2019, 01:39
jfranciscocuencag wrote:

Hello EgmatQuantExpert

I was doing the same approach as you but I had a problem with the following:

If its given that one of the roots is 7, why it doesn't mean that x = -7 because if one root is 7, that would mean that:

(x -7) Isn't it? So the sum of the root -2px shouldn't be (-7) + b?

Could you please help me with this?

Thank you in advance!

Kind regards!


Hi,

    If x - 7 is a factor of the quadratic equation, then we can write (x - 7) = 0, which implies, x = 7

    Similarly, if x = 7 is a root of a quadratic equation, then (x - 7) is a factor of the quadratic equation

Regards,
Sandeep
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Re: Question of the week - 33 (A quadratic equation is in the form ......)   [#permalink] 01 Feb 2019, 01:39
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