Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.6 AIME Find the answer

Solve the system of equations: $20=4a^{2}+9b^{2}$ and $20+12ab=(2a+3b)^{2}$ . Find $ab$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solving the system, we find: $\begin{array}{r} 20=4a^{2}+9b^{2} \\ 20+12ab=4a^{2}+12ab+9b^{2} \\ 20+12ab=100 \\ 12ab=80 \\ ab=\frac{20}{3} \end{array}$

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