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

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The roots of the equations x² + a*x + b = 0 and x + a*√x + b = 0 are not identical due to their fundamental differences. By substituting y = √x in the second equation, it transforms into y² + a*y + b = 0, which shares the same roots as the first equation when solved for y. However, the solutions for x must be derived by squaring the results for y, leading to distinct roots for x. The discussion also touches on the need for clarification regarding the expansion of the second equation into simple fractions.

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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
 
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Because they are different equations! In the second equation, you can substitute [itex]y=\sqrt{x}[/itex] and it becomes [itex]y^2+ay+b=0[/itex]. 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 [itex]y[/itex], not [itex]x[/itex]! To find [itex]x[/itex], you have to square the solution you've got, and the solutions are different.
 


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

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