dvk007 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

GIVEN: x = 7 is one of the roots of the equation x² – 2px + m = 0This 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 120We already know that: x² – 2px + m = (x - 7)(x - k)

If we expand the right side we get: x² – 2px +

m = x² – kx - 7x +

7kNow rewrite the right side as follows: x² –

2px +

m = x² –

(k + 7)x +

7kWe can see that

2p = k + 7And we can see that

m = 7kIn 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 = 12When 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 = 17When 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 = 22When we solve this, we get:

p = 11Aha! 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 +

105So, this meets the condition that says

m is divisible by 5 and is less than 120We now know that

p = 11 and

m = 105All 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 = 0Plug 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 = 120What is the value of p + n – m?p + n – m =

11 +

120 -

105= 26

Answer: D

Cheers,

Brent

_________________

Brent Hanneson – Creator of greenlighttestprep.com

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