Question 13
The number of solutions of the equation $$\sqrt{x+5}+\sqrt{3x+4}=\sqrt{12x+1}$$ is
Solution
Squaring once gives $$\sqrt{(x+5)(3x+4)}=4x-4$$, so a valid solution must satisfy $$x \geq 1$$. Squaring again gives $$13x^2-51x-4=0$$, whose roots are $$x=4$$ and $$x=-\frac{1}{13}$$. Only $$x=4$$ satisfies the required condition and the original equation, so there is exactly one solution.
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