Question: Let $a + \sqrt{2b} = 3 - 2\sqrt{2}$ .Find the value of $a - \sqrt{2b}$
What I did: I compared the whole numbers and the irrational numbers in both sides and calculated the answer $3 + 2\sqrt{2}$.
However, I am not very satisfied to do it this way. I could not manage to calculate it with the algebraic formulas.
How to do it?
Assuming that $a$ and $b$ are integer (or rational), we have that $$a-3=2\sqrt2-\sqrt{2b}=\sqrt2(2-\sqrt b)$$ or $$\sqrt b=2-\frac{a-3}{\sqrt2}$$ Squaring, $$b=4-2\sqrt2(a-3)+\frac{(a-3)^2}2$$ That is, $$2\sqrt2(a-3)=4-b+\frac{(a-3)^2}2$$ R.H.S. is rational. Thus, $a-3=0$.