# When the arguments of two roots of a quadratic equation are equal?

Let $az^2+bz+c=0$ be a quadratic equation with complex coefficients $a,b,c$ and roots $z_1, z_2.$

How can I obtain the condition for $$\arg z_1=\arg z_2$$ containing $a,b,c?$
At present I have, Since $$z_1z_2=\dfrac{c}{a}$$ If $\arg z_1=\arg z_2,$ then $$\arg z_1=\arg z_2=\dfrac{1}{2}(\arg c-\arg a)$$ Is there any reference discuss about roots of quadratic equations with complex coefficients?

Here, I have ask a similar question about $|z_1|=|z_2|,$ which has a nice solution.

• You need to divide the roots, to get a real ratio, not multiply them. – Yves Daoust Jan 5 '15 at 13:34
• Note this is the same as asking when there is a real number $\theta$ such that $(a\theta^2)z^2+(b\theta)z+c$ has two real roots. – Colin McLarty Jan 5 '15 at 14:17

The roots $(-b\pm\sqrt{b^2-4ac})/2a$ are a real positive multiples of each other.

For some positive $\mu$, $$-b+\sqrt{b^2-4ac}=\mu(-b-\sqrt{b^2-4ac})$$ $$(\mu-1)^2b^2=(\mu+1)^2(b^2-4ac)$$ $$b^2=\frac{(\mu+1)^2}{\mu}ac$$

The latter function of $\mu$ can take any value not smaller than $4$, and the condition is

$$b^2=\lambda ac,$$with $\lambda\ge4$, which can be expressed as $2\arg b=\arg a+\arg c\land |b|^2\ge4|ac|$.

$arg(c/a) = 2.arg(b/a)$ and $|b|^2 \ge 4|a|.|c|$ seems to work.

Assume the equation has roots of the form required, i.e. $r_1 e^{i\theta}$ and $r_2 e^{i\theta}$, then $(z-r_1 e^{i\theta} )(z-r_2 e^{i\theta} ) = 0 = z^2 - (r_1 + r_2).e^{i\theta} .z + r_1.r_2.e^{2i\theta}$ Equating the coefficients gives

1. $b/a =- (r_1 + r_2).e^{i\theta}$
2. $c/a = r_1.r_2.e^{2i\theta}$

From this it seems (with $a$ non-zero) we can find $r_1$ and $r_2$ as the real roots of $r^2 + |b/a|.r + |c/a| = 0$, provided $|b|^2 \ge 4|a|.|c|$, and clearly $arg(c/a) = 2.arg(b/a)$.

(Note: if $c = 0$ the solution is degenerate: $z_1 = 0, z_2 = -b/a$ You might still regard $z_1 = 0.arg (z_2)$)

(Note 2: in line with the second answer, the condition that $arg(c/a) = 2.arg(b/a)$ is equivalent to to $b/ac$ is real as $b/ac = (b/a)^2. a/c = (r_1 + r_2)^2.e^{2i\theta}. 1/(r_1.r_2). e^{-2i\theta} = (r_1 + r_2)^2/(r_1.r_2)$ and so together with the condition $|b|^2 \ge 4|a|.|c|$ this is equivalent to $b/ac$ is real and $\ge 4$)

If $a = 0$ then the equation has (at most) one solution, and if $c = 0$ then $z_1 = 0$ is one solution for which $\arg z_1$ is not defined. Therefore in the following I am assuming that both $a$ and $c$ are non-zero.

Assume that $a z^2 + b z + c$ has the two solutions $z_1, z_2$ with $\arg z_1 = \arg z_2$. Then $z_2 = \lambda z_1$ with a real number $\lambda > 0$ and

$$a z^2 + b z + c = a (z - z_1)(z-\lambda z_1)$$ gives the equations $$a \lambda z_1^2 = c \, , \quad - a(1+\lambda)z_1 = b \quad .$$ Substituting $z_1$ from the second equation into the first gives $$\frac {b^2}{a c} = \frac {(1+\lambda)^2}{\lambda} = \frac {(1-\lambda)^2}{\lambda} + 4 \ge 4 \quad.$$

So a necessary condition is that $b^2/(ac)$ is positive real and $\ge 4$.

This condition is also sufficient. If $b^2/(ac) \ge 4$ then it is easy to see that $$\frac {(1+\lambda)^2}{\lambda} = \frac {b^2}{a c}$$ has a real solution $\lambda > 0$ and then $$z_1 = -\frac{b}{a(1+\lambda)} \, , \quad z_2 = -\frac{\lambda b}{a(1+\lambda)}$$ are solutions of $a z^2 + b z + c$ with $z_2 = \lambda z_1$ .