Factoring $a^2+b^2$?

Question

I remember there was a way to factor $a^2+b^2$ into something along the lines of $(a+\sqrt{a}+b)(a-\sqrt{a}+b)$ . I tried every combination of pluses and minuses for this form, but I couldn't get back the original.

Answer 1

We have $$ a^2 + b^2 = a^2 + 2ab + b^2 - 2ab\\ = (a + b)^2 - 2ab\\ = \big((a + b) +\sqrt{2ab}\big)\big((a + b) - \sqrt{2ab}\big) $$ This is a trick commonly attributed to Sophie Germain, if memory serves.