Finding general solution of Radical Equation

  • #1
677
16

Main Question or Discussion Point

Before trying to find out the general solution of a radical equation; I would first like to know if it can be found?
For example I have a equation of the form
[itex]
\text{A1}+\text{A2} x + \text{A3}\sqrt{\text{B1}+\text{B2} x+\text{B3} x^{\frac{3}{2}}+\text{B4}\sqrt{x}+\text{B5} x^2}+ \text{A4}\sqrt{\text{C1}+\text{C2} x+\text{C3} x^{\frac{3}{2}}+\text{C4}\sqrt{x}+\text{C5} x^2}=0[/itex]
Can I find x in terms of the Constants A1,A2 etc?
What is the general view on deciding whether a general solution to radical equation exist or not?
I tried searching, but couldn't find out the answer regarding radical equation.
For polynomial equation though, I learned that a general solution doesn't exist for polynomials of degree 5 or higher.
http://en.wikipedia.org/wiki/Abel–Ruffini_theorem
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
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1. Isolate one of the square roots.
2. Square both sides which will leave a single square root.
3. Isolate that square root.
4. Now you will have an equation involving powers of x of 2 3/2, 1, and 1/2. Let y= x1/2 so that you have a polynomial involving y4, y3, y2, and y.

(There cannot be a "general solution for a polynomial equation" of degee greater than 5 involving only powers and roots of the coefficients because they may have solution that cannot be written in terms of roots.)
 
  • #3
677
16
1. Isolate one of the square roots.
2. Square both sides which will leave a single square root.
3. Isolate that square root and Square both sides
4. Now you will have an equation involving powers of x of 2 3/2, 1, and 1/2. Let y= x1/2 so that you have a polynomial involving y4, y3, y2, and y.


(There cannot be a "general solution for a polynomial equation" of degee greater than 5 involving only powers and roots of the coefficients because they may have solution that cannot be written in terms of roots.)
(I made a little amendments). Thanks.
1.[itex]\text{A1}+\text{A2} x + \text{A3}\sqrt{\text{B1}+\text{B2} x+\text{B3} x^{\frac{3}{2}}+\text{B4}\sqrt{x}+\text{B5} x^2}= \text{A4}\sqrt{\text{C1}+\text{C2} x+\text{C3} x^{\frac{3}{2}}+\text{C4}\sqrt{x}+\text{C5} x^2}[/itex]
2.[itex]\left(\text{A1}+\text{A2} x + \text{A3}\sqrt{\text{B1}+\text{B2} x+\text{B3} x^{\frac{3}{2}}+\text{B4}\sqrt{x}+\text{B5} x^2}\right)^2= \left(\text{A4}\sqrt{\text{C1}+\text{C2} x+\text{C3} x^{\frac{3}{2}}+\text{C4}\sqrt{x}+\text{C5} x^2}\right)^2[/itex]
[itex]\left(\text{A1}^2+\text{A3}^2 \text{B1}+\text{A3}^2 \text{B4} \sqrt{x}+2 \text{A1} \text{A2} x+\text{A3}^2 \text{B2} x+\text{A3}^2 \text{B3} x^{3/2}+\text{A2}^2 x^2+\text{A3}^2 \text{B5} x^2+\text{A3} (2 \text{A1}+2 \text{A2} x) \sqrt{\text{B1}+\text{B4} \sqrt{x}+\text{B2} x+\text{B3} x^{3/2}+\text{B5} x^2}\right)=\text{A4}^2 \text{C1}+\text{A4}^2 \text{C4} \sqrt{x}+\text{A4}^2 \text{C2} x+\text{A4}^2 \text{C3} x^{3/2}+\text{A4}^2 \text{C5} x^2[/itex]
3.[itex]\text{A3} (2 \text{A1}+2 \text{A2} x) \sqrt{\text{B1}+\text{B4} \sqrt{x}+\text{B2} x+\text{B3} x^{3/2}+\text{B5} x^2}=\left(\text{A4}^2 \text{C1}+\text{A4}^2 \text{C4} \sqrt{x}+\text{A4}^2 \text{C2} x+\text{A4}^2 \text{C3} x^{3/2}+\text{A4}^2 \text{C5} x^2\right)-\left(\text{A1}^2+\text{A3}^2 \text{B1}+\text{A3}^2 \text{B4} \sqrt{x}+2 \text{A1} \text{A2} x+\text{A3}^2 \text{B2} x+\text{A3}^2 \text{B3} x^{3/2}+\text{A2}^2 x^2+\text{A3}^2 \text{B5} x^2\right)[/itex]
[itex]\left(\text{A3} (2 \text{A1}+2 \text{A2} x) \sqrt{\text{B1}+\text{B4} \sqrt{x}+\text{B2} x+\text{B3} x^{3/2}+\text{B5} x^2}\right)^2=\left(\left(\text{A4}^2 \text{C1}+\text{A4}^2 \text{C4} \sqrt{x}+\text{A4}^2 \text{C2} x+\text{A4}^2 \text{C3} x^{3/2}+\text{A4}^2 \text{C5} x^2\right)-\left(\text{A1}^2+\text{A3}^2 \text{B1}+\text{A3}^2 \text{B4} \sqrt{x}+2 \text{A1} \text{A2} x+\text{A3}^2 \text{B2} x+\text{A3}^2 \text{B3} x^{3/2}+\text{A2}^2 x^2+\text{A3}^2 \text{B5} x^2\right)\right)^2[/itex]
4.[itex]4 \text{A1}^2 \text{A3}^2 \text{B1}+4 \text{A1}^2 \text{A3}^2 \text{B4} \sqrt{x}+8 \text{A1} \text{A2} \text{A3}^2 \text{B1} x+4 \text{A1}^2 \text{A3}^2 \text{B2} x+4 \text{A1}^2 \text{A3}^2 \text{B3} x^{3/2}+8 \text{A1} \text{A2} \text{A3}^2 \text{B4} x^{3/2}+4 \text{A2}^2 \text{A3}^2 \text{B1} x^2+8 \text{A1} \text{A2} \text{A3}^2 \text{B2} x^2+4 \text{A1}^2 \text{A3}^2 \text{B5} x^2+8 \text{A1} \text{A2} \text{A3}^2 \text{B3} x^{5/2}+4 \text{A2}^2 \text{A3}^2 \text{B4} x^{5/2}+4 \text{A2}^2 \text{A3}^2 \text{B2} x^3+8 \text{A1} \text{A2} \text{A3}^2 \text{B5} x^3+4 \text{A2}^2 \text{A3}^2 \text{B3} x^{7/2}+4 \text{A2}^2 \text{A3}^2 \text{B5} x^4=\text{A1}^4+2 \text{A1}^2 \text{A3}^2 \text{B1}+\text{A3}^4 \text{B1}^2-2 \text{A1}^2 \text{A4}^2 \text{C1}-2 \text{A3}^2 \text{A4}^2 \text{B1} \text{C1}+\text{A4}^4 \text{C1}^2+2 \text{A1}^2 \text{A3}^2 \text{B4} \sqrt{x}+2 \text{A3}^4 \text{B1} \text{B4} \sqrt{x}-2 \text{A3}^2 \text{A4}^2 \text{B4} \text{C1} \sqrt{x}-2 \text{A1}^2 \text{A4}^2 \text{C4} \sqrt{x}-2 \text{A3}^2 \text{A4}^2 \text{B1} \text{C4} \sqrt{x}+2 \text{A4}^4 \text{C1} \text{C4} \sqrt{x}+4 \text{A1}^3 \text{A2} x+4 \text{A1} \text{A2} \text{A3}^2 \text{B1} x+2 \text{A1}^2 \text{A3}^2 \text{B2} x+2 \text{A3}^4 \text{B1} \text{B2} x+\text{A3}^4 \text{B4}^2 x-4 \text{A1} \text{A2} \text{A4}^2 \text{C1} x-2 \text{A3}^2 \text{A4}^2 \text{B2} \text{C1} x-2 \text{A1}^2 \text{A4}^2 \text{C2} x-2 \text{A3}^2 \text{A4}^2 \text{B1} \text{C2} x+2 \text{A4}^4 \text{C1} \text{C2} x-2 \text{A3}^2 \text{A4}^2 \text{B4} \text{C4} x+\text{A4}^4 \text{C4}^2 x+2 \text{A1}^2 \text{A3}^2 \text{B3} x^{3/2}+2 \text{A3}^4 \text{B1} \text{B3} x^{3/2}+4 \text{A1} \text{A2} \text{A3}^2 \text{B4} x^{3/2}+2 \text{A3}^4 \text{B2} \text{B4} x^{3/2}-2 \text{A3}^2 \text{A4}^2 \text{B3} \text{C1} x^{3/2}-2 \text{A3}^2 \text{A4}^2 \text{B4} \text{C2} x^{3/2}-2 \text{A1}^2 \text{A4}^2 \text{C3} x^{3/2}-2 \text{A3}^2 \text{A4}^2 \text{B1} \text{C3} x^{3/2}+2 \text{A4}^4 \text{C1} \text{C3} x^{3/2}-4 \text{A1} \text{A2} \text{A4}^2 \text{C4} x^{3/2}-2 \text{A3}^2 \text{A4}^2 \text{B2} \text{C4} x^{3/2}+2 \text{A4}^4 \text{C2} \text{C4} x^{3/2}+6 \text{A1}^2 \text{A2}^2 x^2+2 \text{A2}^2 \text{A3}^2 \text{B1} x^2+4 \text{A1} \text{A2} \text{A3}^2 \text{B2} x^2+\text{A3}^4 \text{B2}^2 x^2+2 \text{A3}^4 \text{B3} \text{B4} x^2+2 \text{A1}^2 \text{A3}^2 \text{B5} x^2+2 \text{A3}^4 \text{B1} \text{B5} x^2-2 \text{A2}^2 \text{A4}^2 \text{C1} x^2-2 \text{A3}^2 \text{A4}^2 \text{B5} \text{C1} x^2-4 \text{A1} \text{A2} \text{A4}^2 \text{C2} x^2-2 \text{A3}^2 \text{A4}^2 \text{B2} \text{C2} x^2+\text{A4}^4 \text{C2}^2 x^2-2 \text{A3}^2 \text{A4}^2 \text{B4} \text{C3} x^2-2 \text{A3}^2 \text{A4}^2 \text{B3} \text{C4} x^2+2 \text{A4}^4 \text{C3} \text{C4} x^2-2 \text{A1}^2 \text{A4}^2 \text{C5} x^2-2 \text{A3}^2 \text{A4}^2 \text{B1} \text{C5} x^2+2 \text{A4}^4 \text{C1} \text{C5} x^2+4 \text{A1} \text{A2} \text{A3}^2 \text{B3} x^{5/2}+2 \text{A3}^4 \text{B2} \text{B3} x^{5/2}+2 \text{A2}^2 \text{A3}^2 \text{B4} x^{5/2}+2 \text{A3}^4 \text{B4} \text{B5} x^{5/2}-2 \text{A3}^2 \text{A4}^2 \text{B3} \text{C2} x^{5/2}-4 \text{A1} \text{A2} \text{A4}^2 \text{C3} x^{5/2}-2 \text{A3}^2 \text{A4}^2 \text{B2} \text{C3} x^{5/2}+2 \text{A4}^4 \text{C2} \text{C3} x^{5/2}-2 \text{A2}^2 \text{A4}^2 \text{C4} x^{5/2}-2 \text{A3}^2 \text{A4}^2 \text{B5} \text{C4} x^{5/2}-2 \text{A3}^2 \text{A4}^2 \text{B4} \text{C5} x^{5/2}+2 \text{A4}^4 \text{C4} \text{C5} x^{5/2}+4 \text{A1} \text{A2}^3 x^3+2 \text{A2}^2 \text{A3}^2 \text{B2} x^3+\text{A3}^4 \text{B3}^2 x^3+4 \text{A1} \text{A2} \text{A3}^2 \text{B5} x^3+2 \text{A3}^4 \text{B2} \text{B5} x^3-2 \text{A2}^2 \text{A4}^2 \text{C2} x^3-2 \text{A3}^2 \text{A4}^2 \text{B5} \text{C2} x^3-2 \text{A3}^2 \text{A4}^2 \text{B3} \text{C3} x^3+\text{A4}^4 \text{C3}^2 x^3-4 \text{A1} \text{A2} \text{A4}^2 \text{C5} x^3-2 \text{A3}^2 \text{A4}^2 \text{B2} \text{C5} x^3+2 \text{A4}^4 \text{C2} \text{C5} x^3+2 \text{A2}^2 \text{A3}^2 \text{B3} x^{7/2}+2 \text{A3}^4 \text{B3} \text{B5} x^{7/2}-2 \text{A2}^2 \text{A4}^2 \text{C3} x^{7/2}-2 \text{A3}^2 \text{A4}^2 \text{B5} \text{C3} x^{7/2}-2 \text{A3}^2 \text{A4}^2 \text{B3} \text{C5} x^{7/2}+2 \text{A4}^4 \text{C3} \text{C5} x^{7/2}+\text{A2}^4 x^4+2 \text{A2}^2 \text{A3}^2 \text{B5} x^4+\text{A3}^4 \text{B5}^2 x^4-2 \text{A2}^2 \text{A4}^2 \text{C5} x^4-2 \text{A3}^2 \text{A4}^2 \text{B5} \text{C5} x^4+\text{A4}^4 \text{C5}^2 x^4[/itex]

But now there are [itex]x^4,x^{\frac{7}{2}},x^3,x^{\frac{5}{2}},x^2,x^{\frac{3}{2}},x,\sqrt{x}[/itex]
If I replace x = y^2 then I will have polynomial of degree 8.
So it appears the equation won't have general solution. (Atleast not in terms of roots and powers as you said). But I wonder in what form I might get the solution, if at all.
 

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