<s>[INST] Applying binomial expansion and using algebra to find the value of $a+b$ Q) In the expansion of
$f(x)=(1 + ax)^4 (1 + bx)^5$
where $a$ and $b$ are positive integers,
the coefficient of $x^2$ is 66.
Evaluate $a+b$.
My working:
After expanding the expression I simplified it and got
$5b^2+10ab+3a^2=33$
After further simplification, I managed to get
$5(a+b)^2-2a^2=33$
but I'm not sure what to do next.
 [/INST] Your result is correct.
If $a,b\in\mathbb Z^{+}$ then we can write,
$$\begin{align}&5b^2+3a^2≥2\sqrt {15}ab\\
\implies &5b^2+3a^2+10ab≥ab \left(10+2\sqrt {15}\right)\\
\implies &33≥\left(10+2\sqrt {15}\right)ab\\
\implies &ab≤\frac{33}{10+2\sqrt {15}}<2\\
\implies &a=b=1.\end{align}$$
This means, the solution doesn't exist.
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