Roots of $x^2+bx+c=0$ with $|b|\leq 1$ and $|c|\leq 1$ have absolute value at most the Golden Ratio. Coincidence?

Question

I noticed a possibly interesting coincidence. Consider the set of quadratics $$ x^2 + bx + c $$ with $|b| \leq 1$ and $|c| \leq 1$. For all $x_0$ such that $$ x_0^2 + bx_0 + c = 0 $$ we have $$ |x_0| \leq \frac{1 + \sqrt{5}}{2} $$ where the term on the right is the golden ratio. This bound is sharp since $$ x^2 - x - 1 = 0 $$ has the golden ratio as a solution.

This inequality can be proved by using the quadratic equation and making the obvious estimates.

Is there anything deeper going on here?

My friend suggested that there might be a story about eigenvalues lurking here. I'm curious to hear Math SE's thoughts. I'm interested in different ways of looking at this problem, generalizations, or even references to other people making this observation.

Answer 1

What is a pair of roots of a quadratic polynomial from the given family? It is a pair of numbers $\alpha,\beta$ such that $|\alpha+\beta|\leq1$ and $|\alpha\beta|\leq1$. Plotting and shading these regions on the real plane, we get the following solution region.

The region containing the origin bounded by blue, green and two black lines is the solution region. Your observation reflects the fact that this region lies in the square formed by the lines $x=\phi, x=-\phi, y = \phi \ \ \&\ \ y = -\phi$, where $\phi$ is the golden ratio.

Answer 2

Indeed for the equation $az^2+bz+c=0$ where $a,b,c\in \mathbb{C}$ such that $|a|=|b|=|c|=r$ (or up to scale setting $r=1$), the roots are bounded by

$$\frac{\sqrt{5}-1}{2} \le |z| \le \frac{\sqrt{5}+1}{2}$$

which appeared as the problem 4994 of the issue December 2007 and solved in March 2008 at the solutions-problem section of School Science and Mathematics Association.

The loci of the roots by varying either one of the $a=e^{i\alpha}$, $b=e^{i\beta}$ or $c=e^{i\gamma}$ are shown below, notice each locus is confined in the annulus $\phi^{-1} \le r \le \phi$.

See also articles below for your further interest.

• Golden bounds for the roots of quadratic equations
• Zeros of polynomials with cyclotomic coefficients