Carcass 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. \(\sqrt{81x^2+25xy+16y^2}\)
APPROACH #1: Test a pair of values
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 = 1If 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
APPROACH #2: Factoring
When we scan the answer choices, we might recognize that answer choice B is in the form of the following
special product :
\((a-b)^2 = a^2 - 2ab + b^2\)Likewise, we have B. \(\sqrt{49x^2−84xy+36y^2}=\sqrt{(7x-6y)^2}=7x-6y\)
Since we're told \(x\) and \(y\) are INTEGERS, we can be certain that \(7x-6y\) is an integer
Answer: B
Cheers,
Brent