Find all values of $ac-bd$ where $a, b, c, d$ are real numbers, $a^2c^2+a^2d^2+b^2c^2+b^2d^2=2017$, and $ad+bc=44$.

Question

Find all values of $ac-bd$ where $a, b, c, d$ are real numbers, $a^2c^2+a^2d^2+b^2c^2+b^2d^2=2017$, and $ad+bc=44$.

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I noticed that $a^2c^2=(ac)^2$ and so on for all the terms in the polynomial. How does that help? Am I close to something. All help is appreciated.

Answer 1

Well, $(ac-bd)^2+(ad+bc)^2=a^2c^2+a^2d^2+b^2c^2+b^2d^2$.