The below question is from a practice PSAT exam. I know the answer is 0, but I could not figure out how to solve it without doing trial and error. Any tips or tricks on how to solve something like this?

(ax + by) (cx - dy)

In the expression above, a, b, c, and d are non-zero
constants and ad = bc. If ac = 18 and bd = 50, what
is the value of the coefficient of the xy term when the expression is multiplied out and the like terms are collected?

When you multiply it out, the xy terms are -adxy and +bcxy. Since ad = bc, when you add them together, the coefficients of that term will equal 0. It's essentially -adxy + adxy or -bcxy + bcxy, depending on how you want to look at it, The rest is superfluous.

Fully multiplied out, you get: acx^2 - adxy + bcxy -bdy^2

Easier than I expected. Just wasn't seeing it. Thanks.

Just for fun
a= 3
b= 5
c= 6
d= 10

My weird brain would not let me answer the question without solving for unknowns

Foil!

PeekingDuck said:

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Foil!

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Easy to see now. My problem was I was trying to do it without paper and I was focused on thinking I needed to solve for each coefficient to calculate the end result when I applied FOIL.