Is there a way to get rid of all square roots in an equation with $4$ square roots?

Question

Say we have a problem with one square root in it; it's easily solvable: $$\sqrt{a}=b$$ $$a=b^2$$ Two: $$\sqrt{a}+\sqrt{b}=c$$ $$a+b+2\sqrt{ab}=c^2$$ And it reduces to a single root problem. Three: $$\sqrt{a}+\sqrt{b}+\sqrt{c}=d$$ $$\sqrt{a}+\sqrt{b}=d-\sqrt{c}$$ $$a+b+2\sqrt{ab}=d^2+c-2d\sqrt{c}$$ And, it's reduces to a double root problem. Four however seems impossible. Why?

There are a few possibilities at the start with how many terms you'll have at either side of the equals sign before you square:

constant = 4 roots --> results in constant = 6 roots

constant+root = 3 roots --> results in constant + root = 3 roots

constant+ 2 roots = 2 roots --> results in constant + 3 roots = root

constant + 3 roots = root --> results in 6 roots = constant

constant + 4 roots = constant --> results in constant = 10 roots

This is why I think it might be impossible (except when $e=0$ of course).

Question: Is it (im)possible? If not is there a maximum amount of roots for which it is possible?

Answer 1

Here is a strategy - suppose the square roots are initially of $a,b,c,d$

First take all the terms involving $\sqrt a$ to one side, and everything else to the other, and square the resulting equation. This leaves a complicated equation with a number of square roots, some of products of terms, but no square roots of $a$ left in any term.

Now do the same with the terms involving $\sqrt b$ (even where multiplied by other square roots) to eliminate that.

After squaring four times (to deal with $\sqrt c$ and $\sqrt d$ as well), you should have no square roots left. Obviously the strategy can be applied to any such expression.

Answer 2

Here is a more explicit version of Mark Benett's answer. You'll quickly realize that those kind of computations are better done by a computer than by human hand, so I appended some PARI/GP code below.

You have

$$ (\sqrt{a}+\sqrt{b}+\sqrt{c}-e)^2=d $$

Expanding yields $x_1+y_1\sqrt{c}=0$ where

$$ \begin{array}{ccc} x_1 &=& a+b+c-d+e^2-2e(\sqrt{a}+\sqrt{b})-2\sqrt{a}{b} \\ y_1 &=& 2(\sqrt{a}+\sqrt{b}-e) \end{array} $$

So $x_1^2-cy_1^2=0$. Expanded, this means that $x_2+y_2\sqrt{c}=0$ where $x_2,y_2$ are long and complicated terms.

Here is the related PARI/GP code :

? tr1(expr)=polrem(expr,sc^2-c,sc)
%2 = (expr)->polrem(expr,sc^2-c,sc)
? tr2(expr)=polrem(expr,sb^2-b,sb)
%3 = (expr)->polrem(expr,sb^2-b,sb)
? tr3(expr)=polrem(expr,sa^2-a,sa)
%4 = (expr)->polrem(expr,sa^2-a,sa)
? 
? tr(expr)=tr3(tr2(tr1(expr)))
%5 = (expr)->tr3(tr2(tr1(expr)))
? 
? tt1=tr((sa+sb+sc-e)^2-d)
%6 = (2*sb + (2*sc - 2*e))*sa + ((2*sc - 2*e)*sb + (-2*e*sc + (e^2 + (-d + (c + (b + a))))))
? x1=polcoeff(tt1,0,sc)
%7 = (2*sb - 2*e)*sa + (-2*e*sb + (e^2 + (-d + (c + (b + a)))))
? y1=polcoeff(tt1,1,sc)
%8 = 2*sa + (2*sb - 2*e)
? tt2=tr(x1^2-c*y1^2)
%9 = ((12*e^2 + (-4*d + (-4*c + (4*b + 4*a))))*sb + (-4*e^3 + (4*d + (4*c + (-12*b - 4*a)))*e))*sa + ((-4*e^3 + (4*d + (4*c + (-4*b - 12*a)))*e)*sb + (e^4 + (-2*d + (-2*c + (6*b + 6*a)))*e^2 + (d^2 + (-2*c + (-2*b - 2*a))*d + (c^2 + (-2*b - 2*a)*c + (b^2 + 6*a*b + a^2)))))
? x2=polcoeff(tt2,0,sb)
%10 = (-4*e^3 + (4*d + (4*c + (-12*b - 4*a)))*e)*sa + (e^4 + (-2*d + (-2*c + (6*b + 6*a)))*e^2 + (d^2 + (-2*c + (-2*b - 2*a))*d + (c^2 + (-2*b - 2*a)*c + (b^2 + 6*a*b + a^2))))
? y2=polcoeff(tt2,1,sb)
%11 = (12*e^2 + (-4*d + (-4*c + (4*b + 4*a))))*sa + (-4*e^3 + (4*d + (4*c + (-4*b - 12*a)))*e)
? tt3=tr(x2^2-b*y2^2)
%12 = (-8*e^7 + (24*d + (24*c + (24*b - 56*a)))*e^5 + (-24*d^2 + (-16*c + (-16*b + 80*a))*d + (-24*c^2 + (-16*b + 80*a)*c + (-24*b^2 + 80*a*b - 56*a^2)))*e^3 + (8*d^3 + (-8*c + (-8*b - 24*a))*d^2 + (-8*c^2 + (80*b - 16*a)*c + (-8*b^2 - 16*a*b + 24*a^2))*d + (8*c^3 + (-8*b - 24*a)*c^2 + (-8*b^2 - 16*a*b + 24*a^2)*c + (8*b^3 - 24*a*b^2 + 24*a^2*b - 8*a^3)))*e)*sa + (e^8 + (-4*d + (-4*c + (-4*b + 28*a)))*e^6 + (6*d^2 + (4*c + (4*b - 60*a))*d + (6*c^2 + (4*b - 60*a)*c + (6*b^2 - 60*a*b + 70*a^2)))*e^4 + (-4*d^3 + (4*c + (4*b + 36*a))*d^2 + (4*c^2 + (-40*b + 24*a)*c + (4*b^2 + 24*a*b - 60*a^2))*d + (-4*c^3 + (4*b + 36*a)*c^2 + (4*b^2 + 24*a*b - 60*a^2)*c + (-4*b^3 + 36*a*b^2 - 60*a^2*b + 28*a^3)))*e^2 + (d^4 + (-4*c + (-4*b - 4*a))*d^3 + (6*c^2 + (4*b + 4*a)*c + (6*b^2 + 4*a*b + 6*a^2))*d^2 + (-4*c^3 + (4*b + 4*a)*c^2 + (4*b^2 - 40*a*b + 4*a^2)*c + (-4*b^3 + 4*a*b^2 + 4*a^2*b - 4*a^3))*d + (c^4 + (-4*b - 4*a)*c^3 + (6*b^2 + 4*a*b + 6*a^2)*c^2 + (-4*b^3 + 4*a*b^2 + 4*a^2*b - 4*a^3)*c + (b^4 - 4*a*b^3 + 6*a^2*b^2 - 4*a^3*b + a^4))))
? x3=polcoeff(tt3,0,sa)
%13 = e^8 + (-4*d + (-4*c + (-4*b + 28*a)))*e^6 + (6*d^2 + (4*c + (4*b - 60*a))*d + (6*c^2 + (4*b - 60*a)*c + (6*b^2 - 60*a*b + 70*a^2)))*e^4 + (-4*d^3 + (4*c + (4*b + 36*a))*d^2 + (4*c^2 + (-40*b + 24*a)*c + (4*b^2 + 24*a*b - 60*a^2))*d + (-4*c^3 + (4*b + 36*a)*c^2 + (4*b^2 + 24*a*b - 60*a^2)*c + (-4*b^3 + 36*a*b^2 - 60*a^2*b + 28*a^3)))*e^2 + (d^4 + (-4*c + (-4*b - 4*a))*d^3 + (6*c^2 + (4*b + 4*a)*c + (6*b^2 + 4*a*b + 6*a^2))*d^2 + (-4*c^3 + (4*b + 4*a)*c^2 + (4*b^2 - 40*a*b + 4*a^2)*c + (-4*b^3 + 4*a*b^2 + 4*a^2*b - 4*a^3))*d + (c^4 + (-4*b - 4*a)*c^3 + (6*b^2 + 4*a*b + 6*a^2)*c^2 + (-4*b^3 + 4*a*b^2 + 4*a^2*b - 4*a^3)*c + (b^4 - 4*a*b^3 + 6*a^2*b^2 - 4*a^3*b + a^4)))
? y3=polcoeff(tt3,1,sa)
%14 = -8*e^7 + (24*d + (24*c + (24*b - 56*a)))*e^5 + (-24*d^2 + (-16*c + (-16*b + 80*a))*d + (-24*c^2 + (-16*b + 80*a)*c + (-24*b^2 + 80*a*b - 56*a^2)))*e^3 + (8*d^3 + (-8*c + (-8*b - 24*a))*d^2 + (-8*c^2 + (80*b - 16*a)*c + (-8*b^2 - 16*a*b + 24*a^2))*d + (8*c^3 + (-8*b - 24*a)*c^2 + (-8*b^2 - 16*a*b + 24*a^2)*c + (8*b^3 - 24*a*b^2 + 24*a^2*b - 8*a^3)))*e
? tt4=tr(x3^2-b*y3^2)
%15 = e^16 + (-8*d + (-8*c + (-72*b + 56*a)))*e^14 + (28*d^2 + (40*c + (424*b - 344*a))*d + (28*c^2 + (424*b - 344*a)*c + (412*b^2 - 1240*a*b + 924*a^2)))*e^12 + (-56*d^3 + (-72*c + (-1032*b + 888*a))*d^2 + (-72*c^2 + (-1584*b + 1232*a)*c + (-1480*b^2 + 5200*a*b - 4040*a^2))*d + (-56*c^3 + (-1032*b + 888*a)*c^2 + (-1480*b^2 + 5200*a*b - 4040*a^2)*c + (-1016*b^3 + 4856*a*b^2 - 8072*a^2*b + 3976*a^3)))*e^10 + (70*d^4 + (40*c + (1320*b - 1240*a))*d^3 + (36*c^2 + (2136*b - 1448*a)*c + (1828*b^2 - 8360*a*b + 6948*a^2))*d^2 + (40*c^3 + (2136*b - 1448*a)*c^2 + (3928*b^2 - 14064*a*b + 10072*a^2)*c + (1832*b^3 - 11176*a*b^2 + 22104*a^2*b - 11992*a^3))*d + (70*c^4 + (1320*b - 1240*a)*c^3 + (1828*b^2 - 8360*a*b + 6948*a^2)*c^2 + (1832*b^3 - 11176*a*b^2 + 22104*a^2*b - 11992*a^3)*c + (1350*b^4 - 8152*a*b^3 + 18980*a^2*b^2 - 18392*a^3*b + 6470*a^4)))*e^8 + (-56*d^5 + (40*c + (-920*b + 1000*a))*d^4 + (16*c^2 + (-1184*b + 352*a)*c + (-752*b^2 + 6240*a*b - 5872*a^2))*d^3 + (16*c^3 + (-912*b + 368*a)*c^2 + (-4624*b^2 + 12704*a*b - 7696*a^2)*c + (-624*b^3 + 7792*a*b^2 - 20624*a^2*b + 12944*a^3))*d^2 + (40*c^4 + (-1184*b + 352*a)*c^3 + (-4624*b^2 + 12704*a*b - 7696*a^2)*c^2 + (-4768*b^3 + 24224*a*b^2 - 36704*a^2*b + 18272*a^3)*c + (-728*b^4 + 7776*a*b^3 - 26384*a^2*b^2 + 30304*a^3*b - 11992*a^4))*d + (-56*c^5 + (-920*b + 1000*a)*c^4 + (-752*b^2 + 6240*a*b - 5872*a^2)*c^3 + (-624*b^3 + 7792*a*b^2 - 20624*a^2*b + 12944*a^3)*c^2 + (-728*b^4 + 7776*a*b^3 - 26384*a^2*b^2 + 30304*a^3*b - 11992*a^4)*c + (-1016*b^5 + 6888*a*b^4 - 18800*a^2*b^3 + 24976*a^3*b^2 - 16024*a^4*b + 3976*a^5)))*e^6 + (28*d^6 + (-72*c + (312*b - 456*a))*d^5 + (36*c^2 + (216*b + 472*a)*c + (-92*b^2 - 1960*a*b + 2468*a^2))*d^4 + (16*c^3 + (-528*b - 16*a)*c^2 + (3312*b^2 - 4192*a*b + 1264*a^2)*c + (-240*b^3 - 1552*a*b^2 + 7152*a^2*b - 5872*a^3))*d^3 + (36*c^4 + (-528*b - 16*a)*c^3 + (3288*b^2 - 3056*a*b + 2264*a^2)*c^2 + (1008*b^3 - 15600*a*b^2 + 18704*a^2*b - 7696*a^3)*c + (-220*b^4 - 272*a*b^3 + 9688*a^2*b^2 - 14608*a^3*b + 6948*a^4))*d^2 + (-72*c^5 + (216*b + 472*a)*c^4 + (3312*b^2 - 4192*a*b + 1264*a^2)*c^3 + (1008*b^3 - 15600*a*b^2 + 18704*a^2*b - 7696*a^3)*c^2 + (3928*b^4 - 15456*a*b^3 + 30224*a^2*b^2 - 27744*a^3*b + 10072*a^4)*c + (-200*b^5 - 1064*a*b^4 + 8688*a^2*b^3 - 17424*a^3*b^2 + 14040*a^4*b - 4040*a^5))*d + (28*c^6 + (312*b - 456*a)*c^5 + (-92*b^2 - 1960*a*b + 2468*a^2)*c^4 + (-240*b^3 - 1552*a*b^2 + 7152*a^2*b - 5872*a^3)*c^3 + (-220*b^4 - 272*a*b^3 + 9688*a^2*b^2 - 14608*a^3*b + 6948*a^4)*c^2 + (-200*b^5 - 1064*a*b^4 + 8688*a^2*b^3 - 17424*a^3*b^2 + 14040*a^4*b - 4040*a^5)*c + (412*b^6 - 2888*a*b^5 + 8356*a^2*b^4 - 12784*a^3*b^3 + 10916*a^4*b^2 - 4936*a^5*b + 924*a^6)))*e^4 + (-8*d^7 + (40*c + (-24*b + 104*a))*d^6 + (-72*c^2 + (-48*b - 304*a)*c + (56*b^2 + 80*a*b - 456*a^2))*d^5 + (40*c^3 + (408*b + 280*a)*c^2 + (-1064*b^2 + 528*a*b + 472*a^2)*c + (104*b^3 + 152*a*b^2 - 488*a^2*b + 1000*a^3))*d^4 + (40*c^4 + (-672*b - 160*a)*c^3 + (1008*b^2 - 608*a*b - 16*a^2)*c^2 + (864*b^3 + 3232*a*b^2 - 3424*a^2*b + 352*a^3)*c + (-216*b^4 - 416*a*b^3 - 784*a^2*b^2 + 1632*a^3*b - 1240*a^4))*d^3 + (-72*c^5 + (408*b + 280*a)*c^4 + (1008*b^2 - 608*a*b - 16*a^2)*c^3 + (-7056*b^3 + 6544*a*b^2 - 3440*a^2*b + 368*a^3)*c^2 + (1624*b^4 + 928*a*b^3 - 7152*a^2*b^2 + 7072*a^3*b - 1448*a^4)*c + (-8*b^5 + 24*a*b^4 - 656*a^2*b^3 + 2160*a^3*b^2 - 2408*a^4*b + 888*a^5))*d^2 + (40*c^6 + (-48*b - 304*a)*c^5 + (-1064*b^2 + 528*a*b + 472*a^2)*c^4 + (864*b^3 + 3232*a*b^2 - 3424*a^2*b + 352*a^3)*c^3 + (1624*b^4 + 928*a*b^3 - 7152*a^2*b^2 + 7072*a^3*b - 1448*a^4)*c^2 + (-1584*b^5 + 4240*a*b^4 - 7008*a^2*b^3 + 8864*a^3*b^2 - 5744*a^4*b + 1232*a^5)*c + (168*b^6 - 432*a*b^5 - 296*a^2*b^4 + 2144*a^3*b^3 - 2856*a^4*b^2 + 1616*a^5*b - 344*a^6))*d + (-8*c^7 + (-24*b + 104*a)*c^6 + (56*b^2 + 80*a*b - 456*a^2)*c^5 + (104*b^3 + 152*a*b^2 - 488*a^2*b + 1000*a^3)*c^4 + (-216*b^4 - 416*a*b^3 - 784*a^2*b^2 + 1632*a^3*b - 1240*a^4)*c^3 + (-8*b^5 + 24*a*b^4 - 656*a^2*b^3 + 2160*a^3*b^2 - 2408*a^4*b + 888*a^5)*c^2 + (168*b^6 - 432*a*b^5 - 296*a^2*b^4 + 2144*a^3*b^3 - 2856*a^4*b^2 + 1616*a^5*b - 344*a^6)*c + (-72*b^7 + 488*a*b^6 - 1416*a^2*b^5 + 2280*a^3*b^4 - 2200*a^4*b^3 + 1272*a^5*b^2 - 408*a^6*b + 56*a^7)))*e^2 + (d^8 + (-8*c + (-8*b - 8*a))*d^7 + (28*c^2 + (40*b + 40*a)*c + (28*b^2 + 40*a*b + 28*a^2))*d^6 + (-56*c^3 + (-72*b - 72*a)*c^2 + (-72*b^2 - 176*a*b - 72*a^2)*c + (-56*b^3 - 72*a*b^2 - 72*a^2*b - 56*a^3))*d^5 + (70*c^4 + (40*b + 40*a)*c^3 + (36*b^2 + 344*a*b + 36*a^2)*c^2 + (40*b^3 + 344*a*b^2 + 344*a^2*b + 40*a^3)*c + (70*b^4 + 40*a*b^3 + 36*a^2*b^2 + 40*a^3*b + 70*a^4))*d^4 + (-56*c^5 + (40*b + 40*a)*c^4 + (16*b^2 - 416*a*b + 16*a^2)*c^3 + (16*b^3 - 272*a*b^2 - 272*a^2*b + 16*a^3)*c^2 + (40*b^4 - 416*a*b^3 - 272*a^2*b^2 - 416*a^3*b + 40*a^4)*c + (-56*b^5 + 40*a*b^4 + 16*a^2*b^3 + 16*a^3*b^2 + 40*a^4*b - 56*a^5))*d^3 + (28*c^6 + (-72*b - 72*a)*c^5 + (36*b^2 + 344*a*b + 36*a^2)*c^4 + (16*b^3 - 272*a*b^2 - 272*a^2*b + 16*a^3)*c^3 + (36*b^4 - 272*a*b^3 + 2008*a^2*b^2 - 272*a^3*b + 36*a^4)*c^2 + (-72*b^5 + 344*a*b^4 - 272*a^2*b^3 - 272*a^3*b^2 + 344*a^4*b - 72*a^5)*c + (28*b^6 - 72*a*b^5 + 36*a^2*b^4 + 16*a^3*b^3 + 36*a^4*b^2 - 72*a^5*b + 28*a^6))*d^2 + (-8*c^7 + (40*b + 40*a)*c^6 + (-72*b^2 - 176*a*b - 72*a^2)*c^5 + (40*b^3 + 344*a*b^2 + 344*a^2*b + 40*a^3)*c^4 + (40*b^4 - 416*a*b^3 - 272*a^2*b^2 - 416*a^3*b + 40*a^4)*c^3 + (-72*b^5 + 344*a*b^4 - 272*a^2*b^3 - 272*a^3*b^2 + 344*a^4*b - 72*a^5)*c^2 + (40*b^6 - 176*a*b^5 + 344*a^2*b^4 - 416*a^3*b^3 + 344*a^4*b^2 - 176*a^5*b + 40*a^6)*c + (-8*b^7 + 40*a*b^6 - 72*a^2*b^5 + 40*a^3*b^4 + 40*a^4*b^3 - 72*a^5*b^2 + 40*a^6*b - 8*a^7))*d + (c^8 + (-8*b - 8*a)*c^7 + (28*b^2 + 40*a*b + 28*a^2)*c^6 + (-56*b^3 - 72*a*b^2 - 72*a^2*b - 56*a^3)*c^5 + (70*b^4 + 40*a*b^3 + 36*a^2*b^2 + 40*a^3*b + 70*a^4)*c^4 + (-56*b^5 + 40*a*b^4 + 16*a^2*b^3 + 16*a^3*b^2 + 40*a^4*b - 56*a^5)*c^3 + (28*b^6 - 72*a*b^5 + 36*a^2*b^4 + 16*a^3*b^3 + 36*a^4*b^2 - 72*a^5*b + 28*a^6)*c^2 + (-8*b^7 + 40*a*b^6 - 72*a^2*b^5 + 40*a^3*b^4 + 40*a^4*b^3 - 72*a^5*b^2 + 40*a^6*b - 8*a^7)*c + (b^8 - 8*a*b^7 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2 - 8*a^7*b + a^8)))
?