Some stuff I did on an equation with 2 radicals in it. 

 

eqn1 := sqrt(`+`(`*`(2, `*`(x)), 5)) = `+`(9, `-`(sqrt(`+`(x, 6)))) 

`*`(`^`(`+`(`*`(2, `*`(x)), 5), `/`(1, 2))) = `+`(9, `-`(`*`(`^`(`+`(x, 6), `/`(1, 2))))) (1)
 

solve(eqn1) 

10 (2)
 

eqn2 := `+`(`*`(`^`(x, 2)), `-`(`*`(97, `*`(x))), `-`(422)) 

`+`(`*`(`^`(x, 2)), `-`(`*`(97, `*`(x))), `-`(422)) (3)
 

solve(eqn2) 

`+`(`/`(97, 2), `*`(`/`(9, 2), `*`(`^`(137, `/`(1, 2))))), `+`(`/`(97, 2), `-`(`*`(`/`(9, 2), `*`(`^`(137, `/`(1, 2)))))) (4)
 

expand(`*`(`^`(`+`(9, `-`(sqrt(`+`(x, 6)))), 2))) 

`+`(87, `-`(`*`(18, `*`(`^`(`+`(x, 6), `/`(1, 2))))), x) (5)
 

`+`(`+`(`+`(`*`(2, `*`(x)), 5), -87), `-`(x)) 

`+`(x, `-`(82)) (6)
 

eqn3 := % = `+`(`-`(`*`(18, `*`(sqrt(`+`(x, 6)))))) 

`+`(x, `-`(82)) = `+`(`-`(`*`(18, `*`(`^`(`+`(x, 6), `/`(1, 2)))))) (7)
 

expand(`*`(`^`(`+`(x, `-`(82)), 2))) = `^`(`+`(`-`(`*`(18, `*`(sqrt(`+`(x, 6)))))), 2) 

`+`(`*`(`^`(x, 2)), `-`(`*`(164, `*`(x))), 6724) = `+`(`*`(324, `*`(x)), 1944) (8)
 

`+`(rhs(%), `-`(lhs(%))) = 0 

`+`(`-`(`*`(`^`(x, 2))), `*`(488, `*`(x)), `-`(4780)) = 0 (9)
 

`+`(`-`(%)) 

`+`(`*`(`^`(x, 2)), `-`(`*`(488, `*`(x))), 4780) = 0 (10)
 

factor(%) 

`*`(`+`(x, `-`(10)), `*`(`+`(x, `-`(478)))) = 0 (11)
 

This gives x = 10 and x = 478, but x = 478 ain't gonna check. 

 

Squaring both sides is casting a net.  You catch all