# Math Help - algebra problem

1. ## algebra problem

How can we show that there are no (x,y) pairs of real numbers that satisfy xy = 3 and (x + y)^2 = 10 ? Intuitively it's obvious but I'm not sure exactly how to show it..

2. Originally Posted by sarahh
How can we show that there are no (x,y) pairs of real numbers that satisfy xy = 3 and (x + y)^2 = 10 ? Intuitively it's obvious but I'm not sure exactly how to show it..
Brute force:

Sub y = 3/x into (x + y)^2 = 10.

After expanding and simplifying you get $x^2 + \frac{9}{x^2} = 4$. Multiply through by x^2 and re-arrange:

$x^4 - 4x^2 + 9 = 0$

$\Rightarrow (x^2 - 2)^2 - 4 + 9 = 0$

$\Rightarrow (x^2 - 2)^2 = -5$.

Therefore no real solutions.