Problem Solving Question? Sum of the squares

Question

The sum of the squares of two numbers is 247 and the product of the two numbers is 21. How would I find all possible values for the sum of the two numbers?

Answer 1

$$ a^2+b^2 = 247, \space ab=21 \\ (a+b)^2 = a^2+b^2+2ab $$ Just substitute the appropriate values.

Answer 2

$$x^2 + y^2 = 247$$ $$xy = 21 \implies 2xy=42$$

$$x^2 + y^2 +2xy= 289$$

$$x^2 + y^2 -2xy= 205 \text{ (really bad by the way, dirty square root)}$$

Without loss of generality assume $x>y$ and thus $x-y = \sqrt{205}$. Now $x+y = \pm 17$. Solve.