Problem Solving

M7MBA wrote:For integers x and y, which of the following MUST be an integer?

\( A. \ \sqrt{25x^2+30xy+36y^2} \)
\( B. \ \sqrt{49x^2−84xy+36y^2} \)
\( C. \ \sqrt{16x^2−y^2} \)
\( D. \ \sqrt{64x^2−64xy−64y^2} \)
\( E. \ \sqrt{81x^2+25xy+16y^2} \)

The OA is the option _B_

Source: Veritas Prep

One approach:

The question is asking us to determine which expression MUST be an integer for ALL integer values of x and y.
So, let's TEST a pair of values.
Let's plug in x = 1 and y = 1
If an expression evaluates to be a non-integer, we can ELIMINATE that answer choice.

We get...
A)√91.This does NOT evaluate to be an integer. ELIMINATE C
B)√1 = 1. This IS an integer. So, keep B
C)√15. This does NOT evaluate to be an integer. ELIMINATE C
D)√-64. Cannot evaluate. ELIMINATE D
E)√122. This does NOT evaluate to be an integer. ELIMINATE E

By the process of elimination, the correct answer is B

Cheers,
Brent