# Finding general solution of Radical Equation

## 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
$\text{A1}+\text{A2} x + \text{A3}\sqrt{\text{B1}+\text{B2} x+\text{B3} x^{\frac{3}{2}}+\text{B4}\sqrt{x}+\text{B5} x^2}+ \text{A4}\sqrt{\text{C1}+\text{C2} x+\text{C3} x^{\frac{3}{2}}+\text{C4}\sqrt{x}+\text{C5} x^2}=0$
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

HallsofIvy
Homework Helper
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.)

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.$\text{A1}+\text{A2} x + \text{A3}\sqrt{\text{B1}+\text{B2} x+\text{B3} x^{\frac{3}{2}}+\text{B4}\sqrt{x}+\text{B5} x^2}= \text{A4}\sqrt{\text{C1}+\text{C2} x+\text{C3} x^{\frac{3}{2}}+\text{C4}\sqrt{x}+\text{C5} x^2}$
2.$\left(\text{A1}+\text{A2} x + \text{A3}\sqrt{\text{B1}+\text{B2} x+\text{B3} x^{\frac{3}{2}}+\text{B4}\sqrt{x}+\text{B5} x^2}\right)^2= \left(\text{A4}\sqrt{\text{C1}+\text{C2} x+\text{C3} x^{\frac{3}{2}}+\text{C4}\sqrt{x}+\text{C5} x^2}\right)^2$
$\left(\text{A1}^2+\text{A3}^2 \text{B1}+\text{A3}^2 \text{B4} \sqrt{x}+2 \text{A1} \text{A2} x+\text{A3}^2 \text{B2} x+\text{A3}^2 \text{B3} x^{3/2}+\text{A2}^2 x^2+\text{A3}^2 \text{B5} x^2+\text{A3} (2 \text{A1}+2 \text{A2} x) \sqrt{\text{B1}+\text{B4} \sqrt{x}+\text{B2} x+\text{B3} x^{3/2}+\text{B5} x^2}\right)=\text{A4}^2 \text{C1}+\text{A4}^2 \text{C4} \sqrt{x}+\text{A4}^2 \text{C2} x+\text{A4}^2 \text{C3} x^{3/2}+\text{A4}^2 \text{C5} x^2$
3.$\text{A3} (2 \text{A1}+2 \text{A2} x) \sqrt{\text{B1}+\text{B4} \sqrt{x}+\text{B2} x+\text{B3} x^{3/2}+\text{B5} x^2}=\left(\text{A4}^2 \text{C1}+\text{A4}^2 \text{C4} \sqrt{x}+\text{A4}^2 \text{C2} x+\text{A4}^2 \text{C3} x^{3/2}+\text{A4}^2 \text{C5} x^2\right)-\left(\text{A1}^2+\text{A3}^2 \text{B1}+\text{A3}^2 \text{B4} \sqrt{x}+2 \text{A1} \text{A2} x+\text{A3}^2 \text{B2} x+\text{A3}^2 \text{B3} x^{3/2}+\text{A2}^2 x^2+\text{A3}^2 \text{B5} x^2\right)$
$\left(\text{A3} (2 \text{A1}+2 \text{A2} x) \sqrt{\text{B1}+\text{B4} \sqrt{x}+\text{B2} x+\text{B3} x^{3/2}+\text{B5} x^2}\right)^2=\left(\left(\text{A4}^2 \text{C1}+\text{A4}^2 \text{C4} \sqrt{x}+\text{A4}^2 \text{C2} x+\text{A4}^2 \text{C3} x^{3/2}+\text{A4}^2 \text{C5} x^2\right)-\left(\text{A1}^2+\text{A3}^2 \text{B1}+\text{A3}^2 \text{B4} \sqrt{x}+2 \text{A1} \text{A2} x+\text{A3}^2 \text{B2} x+\text{A3}^2 \text{B3} x^{3/2}+\text{A2}^2 x^2+\text{A3}^2 \text{B5} x^2\right)\right)^2$
4.$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$

But now there are $x^4,x^{\frac{7}{2}},x^3,x^{\frac{5}{2}},x^2,x^{\frac{3}{2}},x,\sqrt{x}$
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.