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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 \(120.\)

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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 \(120.\)

by VJesus12 » Fri Oct 16, 2020 1:09 am

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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

Answer: D

Source: e-GMAT
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Re: A quadratic equation is in the form of \(x^2-2px + m = 0,\) where \(m\) is divisible by \(5\) and is less than \(120

by deloitte247 » Fri Oct 23, 2020 9:35 pm
$$x^2-2px+m=0$$
m is divisible by 5 and less 102 and has root as 7
$$7^2-2p\left(7\right)+m=0$$
$$49-14p+m=0$$
m = 14p - 49
Since m is divisible by 5, 14p - 49 is a multiple of 5. Also, since p is a prime number, the only prime that can make m divisible by 5 and less than 120 is 11
Therefore, m = 14 (11) - 49
m = 154 - 49 = 105
$$For\ x^2-2px+n=0$$
root is 12
$$12^2-2p\left(12\right)+n=0\ \ \ where\ p=11$$
$$144-2\left(11\right)\left(12\right)+n=0$$
144 - 264 + n = 0
n = 120

Therefore, p + n - m = 11 + 120 - 105 = 26
Answer = option D
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