<s>[INST] Cannot solve by hand:$ x + y = 2; 4x^2 + y^2 = 5(2x - y)(xy)^{\frac12}$ Firstly, this is not my homework. I am well past high school (finished graduate school several years ago) but I am mentoring a high schooler and I want to explain how to solve this by hand using just pen and paper.
A more presentable form + solution is here
My own attempts could only simplify it to a fourth degree polynomial in $x$ and equate it to $0$. There's gotta be a better way but it is beyond me and I feel embarrassed!
 [/INST] since $xy>0,x+y=2>0, \implies x>0,y>0$
let $x=a^2,y=b^2,a>0,b>0 \implies 4a^4+b^4=5(2a^2-b^2)ab \iff \\ 4a^4-10a^3b+5ab^3+b^4=0$
by observation, $a=b$ is a solution as $4-10+5+1=0$
so we get $(a-b)(4a^3-6a^2b-6ab^2-b^3)=0$
now to check  $4a^3-6a^2b-6ab^2-b^3=0$,a factor $b=-2a$ is existed. $\implies$
$(2a+b)(2a^2-4ab-b^2)=0 \implies 2a^2-4ab-b^2=0 $
with $a^2+b^2=2 $ we get $3a^2-4ab=2 \implies b=\dfrac{3a^2-2}{4a} \implies x+\dfrac{(3x-2)^2}{16x}=2$ 
$25x^2-44x+4=0$
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