Solving the equation $\sqrt{x+2} + \sqrt{4x+1} = 5$.

Question

Find the set of all the solutions of the equation $$\sqrt{x+2} + \sqrt{4x+1} = 5$$

I got two solutions using the quadratic formula: $238/9$ and $2$, but only $2$ works in the original equation, so the other solution is extraneous.

Are these the solutions if you use the quadratic formula, and is there a way of solving the equation without squaring the radicals?

Answer 1

Hint

Let $\sqrt{x+2}=a\ge0$

and $\sqrt{4x+1}=b\ge0$

Now $a+b=5$

and $b^2-4a^2=4x+1-4(x+2)$

$(5-a)^2-4a^2=-7$

Can you take it from here?

Answer 2

Without taking squares. By inspection, $2$ is a solution. Notice that both square roots are increasing functions so the LHS is an increasing function for $x\ge -1/4$. For smaller $x$ the LHS is not defined. Thus the equation can have only one solution, $x=2$ (an increasing function takes any value at most once).