PSAT #2, Section 3, #17

There are a lot of words in this one, but all you’re really being asked to do is FOIL and then substitute. Here’s the FOIL part:

    \begin{align*}(ax+by)(cx-dy)\\=acx^2-adxy+bcxy-bdy^2\\=acx^2-(ad-bc)xy-bdy^2\end{align*}

From there, the only given information you need to use is that ad = bc. Because that’s true, the xy coefficient simplifies to zero! Since you’re only asked for the value of that coefficient, grid in zero and move on.

    \begin{align*}(ax+by)(cx-dy)\\=acx^2-adxy+bcxy-bdy^2\\=acx^2-(ad-bc)xy-bdy^2\\=acx^2-0xy-bdy^2\\=acx^2-bdy^2\end{align*}

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