How can I make a square from these terms?

Question

Can anyone help me to make a single square from these terms below:

$$\frac{(x-a)^2}{A} + \frac{(x-b)^2}{B} = \frac{1}{AB}\left((A+B)x^2 - 2(aB + bA)x + a^2B + b^2A\right)$$

Thanks.

Answer 1

$$ \frac{(A+B)x^2 - 2(aB + bA)x + a^2B + b^2A}{AB}=\frac{A+B}{AB}\Big(x^2-2\frac{aB+bA}{A+B}x+\frac{a^2B + b^2A}{A+B}\Big)$$ Now $$x^2-2\frac{aB+bA}{A+B}x+\frac{a^2B + b^2A}{A+B}=\Big(x-\frac{aB+bA}{A+B}\Big)^2-\Big(\frac{aB+bA}{A+B}\Big)^2+\frac{a^2B + b^2A}{A+B}$$Simplify the two last terms to get $$\Big(x-\frac{aB+bA}{A+B}\Big)^2+\frac{A B (a-b)^2}{(A+B)^2}$$ Multiply the result by $\frac{A+B}{AB}$.

Answer 2

You can complete squares as $\frac{1}{AB}(A(x-b)^2+B(x-a)^2)$