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

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The roots of the equations x^2 + a*x + b = 0 and x + a*Sqrt[x] + b = 0 differ because they represent different mathematical forms. The second equation can be transformed by substituting y = Sqrt[x], resulting in the quadratic equation y^2 + a*y + b = 0. While the solutions for y correspond to those of the first equation, they must be squared to find the values of x, leading to different roots. The discussion also seeks clarification on how to express the second equation in simple fractions, indicating a need for further explanation on that topic. Understanding these distinctions is crucial for solving the equations accurately.
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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 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.
 


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

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