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

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In summary, the conversation discusses the differences between two equations and how to expand the second equation in simple fractions. The possibility of factoring the equation is also mentioned.
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Lucasss84
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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]
 
  • #3


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.
 

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

Why are the roots of equations x^2 + a*x + b = 0 and x + a*Sqrt[x] + b = 0 different?

The two equations have different forms and thus have different methods for solving them. The first equation is a quadratic equation, which can be solved using the quadratic formula. The second equation is a radical equation, which requires isolating the square root term and then squaring both sides to eliminate the radical.

Can the roots of these equations be complex numbers?

Yes, the roots of both equations can be complex numbers. This occurs when the discriminant, b^2 - 4ac, is negative in the quadratic equation, or when the radicand, x, is negative in the radical equation.

How do you find the roots of these equations?

The roots of the quadratic equation can be found using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. The roots of the radical equation can be found by isolating the square root term and then squaring both sides to eliminate the radical.

What is the significance of the coefficients a and b in these equations?

The coefficient a represents the coefficient of the x^2 term in the quadratic equation, and the coefficient of the square root term in the radical equation. It affects the shape and position of the graph of the equation. The coefficient b represents the constant term, which affects the y-intercept of the graph.

Are there any special cases for these equations?

Yes, there are special cases for both equations. In the quadratic equation, if the coefficient a is equal to 0, then the equation becomes a linear equation and the solution is a single root. In the radical equation, if the coefficient a is equal to 0, then the equation becomes a pure quadratic equation and can be solved using the quadratic formula.

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