Why the roots of Eq. x^2 + a*x + b = 0 and of Eq. x + a*Sqrt[x] + b =

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The discussion centers on the differences between the roots of the equations x² + a*x + b = 0 and x + a*√x + b = 0. Participants clarify that these equations are inherently different, thus their roots cannot be identical. To expand the second equation into simple fractions, one participant suggests substituting y = √x to find the roots, which can then be factored as (√x - ξ₁)(√x - ξ₂). This method provides a pathway to simplify the equation.

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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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Lucasss84 said:
Why the roots of Eq. x^2 + a*x + b = 0 and of Eq. x + a*Sqrt[x] + b = 0 are not identically?

Why would they be identical? They are different equations, so I see no reason why they should be identical.

How can I expand the second Eq. in simple fractions: x + a*Sqrt[x] + b = ... ?

What do you mean with "simple fractions"??

Do you mean that you want to factor it? Well, first you need to find the roots, you can do that by substituting [itex]y=\sqrt{x}[/itex].
Once you found the roots [itex]\xi_1,\xi_2[/itex], then we can factor

[tex](\sqrt{x}-\xi_1)(\sqrt{x}-\xi_2)[/tex]
 


I'm no mathmetician, but seems like you don't really need a number, just a variable. Look up the imaginary number i. Wish I was more educated.
 

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