how to solve $\sqrt{2x+5}+\sqrt{5x+6}=\sqrt{12x+25}$

Question

Again a root problem.. $\sqrt{2x+5}+\sqrt{5x+6}=\sqrt{12x+25}$

Isn't there any standardized way to solve root problems..Can u plz help by giving some tips and stategies for root problems??

Answer 1

Square to get $$2x+5+2\sqrt{(2x+5)(5x+6)}+5x+6=12x+25$$

This reduces to $$2\sqrt{(2x+5)(5x+6)}=5x+14$$

Now square again, solve the quadratic, and check the solutions in the original equation. It doesn't get that unwieldy, and there is a solution hidden quite close to the surface - I found the formulation of the problem suggestive.

Answer 2

There isn't much you can do which is applicable to every problem of the form $$\sqrt{ax + b} + \sqrt{cx + d} = \sqrt{ex + f} $$ But, if you are lucky enough to have an equation where there exists $p,q$ such that$(ax+b)*(cx+d) = (qx + p)^2$ then this problem reduces quite nicely. You'd find that $$ex + f = (a + c + 2q)x + (b + d + 2p)$$

Of course, extraneous solutions will need to be considered, but .. I think you get my point