BTGModeratorVI wrote: ↑Sun Aug 02, 2020 7:01 am
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
GIVEN: x = 7 is one of the roots of the equation x² – 2px + m = 0
This means (x - 7) must be one of the factors of the expression on the left side of the equation.
That is, x² – 2px + m = 0, can be rewritten as (x - 7)(x +/- something) = 0
[notice that x = 7 is definitely a solution to the new equation]
Let's assign the variable k to the missing number (aka "something")
We can write: x² – 2px + m = (x - 7)(x - k)
GIVEN: m is divisible by 5 and is less than 120
We already know that: x² – 2px + m = (x - 7)(x - k)
If we expand the right side we get: x² – 2px +
m = x² – kx - 7x +
7k
Now rewrite the right side as follows: x² –
2px +
m = x² –
(k + 7)x +
7k
We can see that
2p = k + 7
And we can see that
m = 7k
In order for m to be divisible by 5, it must be the case that k is divisible by 5.
So, k COULD equal 5, 10, 15, 20, 25, etc
Let's test a few possible values of k
If k = 5, then
2p = 5 + 7 = 12
When we solve this, we get: p = 6
HOWEVER, we're told that p is PRIME
So, it cannot be the case that k = 5
If k = 10, then
2p = 10 + 7 = 17
When we solve this, we get: p = 8.5
HOWEVER, we're told that p is PRIME
So, it cannot be the case that k = 10
If k =
15, then
2p = 15 + 7 = 22
When we solve this, we get:
p = 11
Aha! 11 is PRIME
So, it COULD be the case that k = 15. Let's confirm that this satisfies the other conditions in the question.
If k =
15, then we get: x² – 2px + m = (x - 7)(x -
15)
Expand and simplify the right side: x² –
2px +
m = x² –
22x +
105
So, this meets the condition that says
m is divisible by 5 and is less than 120
We now know that
p = 11 and
m = 105
All we need to do now is determine the value of n
GIVEN: x = 12 is one of the solutions of the equation x² – 2px + n = 0
Plug in x = 12 to get: 12² – 2p(12) + n = 0
Since we already know that
p = 11, we can replace p with 11 to get: 12² – 2(11)(12) + n = 0
Simplify: 144 - 264 + n = 0
Simplify: -120 + n = 0
Solve:
n = 120
What is the value of p + n – m?
p + n – m =
11 +
120 -
105
= 26
Answer: D
Cheers,
Brent
