<s>[INST] Solution to $\sqrt{x-2} = 3- 2\sqrt{ x}$ The above question is from Serge Lang's basic mathematics. The question asks if there are any values of x which satisfy the above equation.
Serge Lang's answer key states that there is no solution.
From the equation, I squared both sides, then used the quadratic formula to find $\sqrt{ x} = 2-\sqrt{1/3}$ or $2+\sqrt{1/3}$.
I substituted the former into the equation, and found that it would equate $\sqrt{x-2}$ to a negative answer, and hence is wrong since $\sqrt{x-2}$ refers to the positive root of $x-2$.
For the second answer, I substituted and found that both sides of the equation equaled out. I drew $y=\sqrt{x-2}$ and $y=\sqrt{x}$ in a graphing calculator, and found they intersected correctly at a point.
Yet the book states there are no solutions. What went wrong?
 [/INST] $$\sqrt{x-2}+2\sqrt x=3$$
$x\ge2$
$$x-2+4\sqrt {x(x-2)}+4x=9$$
$$4\sqrt {x(x-2)}=11-5x$$
$x\le\frac{11}5$
$$16x(x-2)=121-110x+25x^2$$
$$9x^2-78x+121=0$$
$$x_{1,2}=\frac{13}{3}\pm\frac{4}{\sqrt3}$$
But $x_{1}=\frac{13}{3}+\frac{4}{\sqrt3}>\frac{11}5$
$$2<x_2=\frac{13}{3}+\frac{4}{\sqrt3}<\frac{11}5$$
Then $x_2=\frac{13}{3}-\frac{4}{\sqrt3} -$ solution of the $\sqrt{x-2}+2\sqrt x=3$
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