# finding complex zeros of a square root algebraic expression

FORENOTE: My question relates to finding the eigenvalues of a non linear map evaulauted at a fixed point in order to solve for the bifurcations of the dynamical system wrt a parameter c, although I am interested purely in the mathematics of my question.

The expression I am trying to solve (for c) is as follows:
$$\frac{9 + 4c \pm{\sqrt{225 + 48c + 16c^2}}}{12} = 1$$
This can be simplified to a root finding problem
$$-3 + 4c \pm{\sqrt{225 + 48c + 16c^2}} = 0$$

To my understanding there should be 2 or maybe even 4 answers (complex answers included) for c however I am only able to find 1. The answer I am able to find is somewhat trivial and comes from rearranging the above equation and squaring both sides to find c = -3:
$$225 + 48c + 16c^2 = (3-4c)^2 \therefore c = -3$$

At the very least it seems that there should be two answers for c, one for the $$+$$ case of $$\pm$$ and the other for the $$-$$ case, but I do not know how to find them. Indeed there should be two answers because c = -3 is only valid for the $$+$$ case (upon evaluating the 2nd expression for the $$-$$ case the answer is $$-30$$ which is not $$0$$ i.e. roots are not repeated). Intuition also tells me that, since there is a quadratic involved in the square root, each of these cases may even have two solutions for c.
In essence I am looking for a mathematical way of finding the other answer(s) for c, or an explanation as to why it does not exist.

• Thats the game isn't it... it wouldn't be mathematics if we were only interested in real solutions :) Commented Jul 22, 2022 at 13:33
• See WolframsAlpha's take on this. Commented Jul 22, 2022 at 16:17
• @MatthewEdizBeadman "there should be two answers for $c$" $\;-\;$ No, there is only one root, and you just proved that. Squaring an equation can introduce extraneous roots, but it never discards legitimate roots of the original equation.
– dxiv
Commented Jul 22, 2022 at 20:50
• It may be clearer to work this backwards: start with $m=0$. Add some quantity $x^2$, so $x^2+m = x^2$. This has an infinity of solutions for $x$. Now take a square root: $\sqrt{x^2+m} = x$. This has the same number of solutions. $x^2+m=x^2$ is not a quadratic equation, and the only conclusion is that $m=0$. Commented Jul 23, 2022 at 11:53
• Thank you all for your help, I guess I couldn't convince myself that there could be no answer for the - case ($-c+4c-\sqrt{225+48c+16c^2}=0$), but im happy to move on now :) Commented Jul 23, 2022 at 14:56

$$225+48x+16 x^2 - (3-4x)^2=0$$ has no term in $$x^2$$, the two terms cancel each other out, leaving you with a linear equation in $$x$$.

A numeric search for a complex solution for the negative case came up with nothing:

package main

import (
"fmt"
"math"
"math/cmplx"
)

func main() {
limit := 1000.0
step := 0.1
min := math.Inf(1)
for x := -limit; x <= limit; x += step {
for y := -limit; y <= limit; y += step {
c := complex(x, y)
ex := complex(3, 0) + 4*c - cmplx.Sqrt(complex(225, 0)+48*c+16*c*c)
a := cmplx.Abs(ex)
if a < min {
if a < 100 {
fmt.Printf("%g %v %v\n", a, c, ex)
}
min = a
}
}
}
}


The minimum this achieved on my computer is this:

3.0000930225447293 (-1.4999999998411289-1000i) (-3.000000000000004-0.023625069768058893i)


This is Google Go language, you can run it in the browser here: https://go.dev/play/p/yEYdLxdm9on However, it will time out, so you will need to change the limits.

• indeed, but if you try to input that solution into the minus case it does not give you the correct answer of 1 (relating to the first equation) and therefore the solution is incomplete. I know that the second solution will be complex, I just dont know how to find it. In a sense information is lost when you square both sides, and I am curious how to regain that information. Commented Jul 22, 2022 at 13:30
• @MatthewEdizBeadman a numeric search for a complex solution from -1000 to 1000 comes up with nothing whatsoever, and it seems to reach a minimum of $-3+0i$ as the imaginary part goes to infinity. I've added the code here so you can check yourself. Commented Jul 22, 2022 at 22:19