Why the roots of Eq. x^2 + a*x + b = 0

[Lucasss84]

Why the roots of Eq. x^2 + a*x + b = 0 and of Eq. x + a*Sqrt[x] + b = 0 are not identically? How can I expand the second Eq. in simple fractions: x + a*Sqrt[x] + b = ... ?
Thank you. Lucas

 

[dalcde]

Because they are different equations! In the second equation, you can substitute $y=\sqrt{x}$ and it becomes $y^2+ay+b=0$. Now the solution to the second equation is the same as the solution of the first, but remember that the solution we've got for the second equation is $y$, not $x$! To find $x$, you have to square the solution you've got, and the solutions are different.

 

[HallsofIvy]

Now, what do you mean by "expand in simple fractions"?