<s>[INST] Solve $2x^2-5x+2=$ $\frac{5-\sqrt{9+8x}}{4}$ Solve $2x^2-5x+2$= $\frac{5-\sqrt{9+8x}}{4}$
I simply do square both sides solve it and I get two value of x one is 2 and other is $\frac{3-√5}{2}$ but this approach it take more time so is there any approach for solving this equation.
 [/INST] Taking Jerry Chang's observation (which amounts to the fact that the expression on the right-hand side is what you get when you plug the coefficients of the quadratic into the quadratic formula, just choosing the minus sign for the square root) and setting $y=2x^2-5x+2$ so that $x=2y^2-5y+2$ we can subtract one of these from the other to obtain:
$$y-x=2(x^2-y^2)-5(x-y)$$
Which yields $y=x$ ; or
$1=5-2(x+y)$ ie $x+y=2$
Then the problem splits as $2x^2-5x+2=x$ or $2x^2-5x+2=2-x$
The solutions of these equations have to be plugged back into the original for checking to see which belongs to which choice of sign of the square root.
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