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Question
Write the following square of binomial as trinomial: \[\left( \frac{2a}{3b} + \frac{2b}{3a} \right)^2\]
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Solution
We will use the identities
\[\left( a + b \right)^2 = a^2 + 2ab + b^2 \text { and } \left( a - b \right)^2 = a^2 - 2ab + b^2\] to convert the squares of binomials as trinomials.
\[ \left( \frac{2a}{3b} + \frac{2b}{3a} \right)^2 \]
\[ = \left( \frac{2a}{3b} \right)^2 + 2$\left( \frac{2a}{3b} \right)$\left( \frac{2b}{3a} \right) + \left( \frac{2b}{3a} \right)^2 \]
\[ = \frac{4 a^2}{9 b^2} + \frac{8}{9} + \frac{4 b^2}{9 a^2}\]
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