# How to prove this equation has no real answer

I wanted to solve this equation , after trying several approaches , wondered if it has a real answer . I used Wolframalpha and it told me that it does not have any ! Can we ever prove that an equation does not have a real answer? how ? $$\sqrt{72 - x} + \sqrt{x - \frac{72}{x}} = x$$ I tried to indicate the common domain , I expected it to be an empty set but it wasn't.

• Reading the title, I thought this was some kind of variation on the Russell paradox. – Marc van Leeuwen Dec 15 '13 at 17:57
• @MarcvanLeeuwen it might be! – amatr Dec 15 '13 at 22:41

If you move the $x$ to the LHS, and if you proved that the LHS function is always greater than zero then you can conclude that there is no solution to this function for real numbers. In this case, I believe it has a solution and its $x=9$

Your new equation does not have a solution. While there is domain for which both of the functions overlap, there is no solution. This is best seen with a graph of the function.

• Oh yes , what a mistake . – amatr Dec 15 '13 at 17:09
• I edited the equation , sorry. – amatr Dec 15 '13 at 17:10
• I think graph isn't a proof. – amatr Dec 15 '13 at 18:13
• You can estimate the graph using calculus and from there you can see that it never intersects the x-axis. But, this process would take so much time. Using a GC program is easier... – John Dec 15 '13 at 18:16

This equation has only one solution and it can check by drawing graph that y = x cut y = sqrt(72 - 72/x) +sqrt(x-72/x) at only one point .

• $\sqrt{63}+1\ne 9$. – Andrés E. Caicedo Dec 15 '13 at 17:24
• Oh You have edited the question . – john Dec 15 '13 at 17:28
• is drawing the graph a strong proof ? isn't that intuitive? – amatr Dec 15 '13 at 18:12
• @ amtr No, drawing graph is not easy all time here also it is difficult to draw graph for L.H.S. function . It is better sometime to draw a graph if proof is not easy or not getting through proof or tired by proof . – john Dec 17 '13 at 16:35