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# Complex #'s

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Let   $$a,b,c, z$$   be complex numbers such that   $$|a| = |b| = |c| > 0$$   and   $$az^2+bz+c=0$$
Find the largest possible value of  $$|z|$$.

Nov 18, 2019

#1
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Let a = b = c.  Then the quadratic is az^2 + az + a = a(z^2 + z + 1) = 0.  The roots of z are -1/2 \pm sqrt(3)/2*i, and these have magnitude 1, so this is the maximum.

Nov 18, 2019
#2
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I don't believe we can assume that a = b = c, since all we are given is that their three moduli are equal and nonzero. So I will provide an answer that does not reqire that assumption.

We are given that $$az^2+bz + c = 0$$, so we can assume

$$0= |az^2+bz+c| \geq|az^2+bz|-|c| \geq|az^2|-|bz|-|c|$$.

But then

$$|a||z|^2-|b||z|-|c|=k|z|^2-k|z|-k=k(|z|^2-|z|-1)\leq0$$,

where $$k=|a|=|b|=|c|$$is nonzero and positive.

This forces the inequality $$|z|^2-|z|-1\leq0$$.

From the graph of the real-valued function $$f(|z|)=|z|^2-|z|-1$$ given above, one can see that the values of $$|z|$$ for the portion of the graph below and touching the $$|z|$$ axis range from

|z|= $$\frac{1-\sqrt{5}}{2}$$ to |z|=$$\frac{1+\sqrt{5}}{2}$$. So the largest value of |z| is $$\frac{1+\sqrt{5}}{2}$$.

Nov 18, 2019