Quadratic equations in C

The real point of complex numbers appears here: in C, every polynomial equation has solutions. Never again "no solution".

Quadratics with negative discriminant

Take az² + bz + c = 0 with real coefficients and Δ = b² - 4ac < 0. In R we stopped there. In C we go on, because Δ now has square roots: if Δ = -36, then √Δ = ± 6i.

z² - 4z + 13 = 0

Δ = 16 - 52 = -36 < 0        √Δ = 6i

        4 ± 6i
z = ------------- = 2 ± 3i
          2

Solutions:  z1 = 2 + 3i   and   z2 = 2 - 3i

The formula is the same as in the real case, with i√|Δ| in place of √Δ.

The solutions are conjugate

This is no accident:

Δ < 0  and REAL coefficients   ->   z1 and z2 are CONJUGATE

In the complex plane the two solutions are symmetric about the real axis:

              ^
          3   +----• 2 + 3i
              |    |
      --------+----+------>
              |    |
         -3   +----• 2 - 3i

This is a free check: if an equation with real coefficients yields two non-conjugate solutions, there is a computational error.

Sum and product, still valid

z1 + z2 = -b/a           (2+3i) + (2-3i) = 4 = 4/1        ✔
z1 × z2 =  c/a           (2+3i)(2-3i) = 4 + 9 = 13        ✔

These relations give the fastest verification.

An example worth knowing

z² + z + 1 = 0        Δ = 1 - 4 = -3        √Δ = i√3

      -1 ± i√3
z = -------------
          2

These two solutions are the cube roots of unity other than 1: cubed, they give 1. They turn up everywhere, from equilateral triangles to three-phase electrical systems.

The fundamental theorem

The result that justifies the whole chapter, proved by Gauss:

Fundamental theorem of algebra. Every non-constant polynomial with complex coefficients has at least one root in C.

By induction it follows that a polynomial of degree n has exactly n roots in C, counted with multiplicity, and factors completely:

P(z) = a (z - z1)(z - z2) ... (z - zn)

We say C is algebraically closed. That is what R lacked: the polynomial x² + 1 had no root there, whereas it has two in C, i and -i.

What it is really for

The usefulness goes far beyond algebra:

Electricity        : complex impedance -> AC circuits are handled with Ohm's
                     law, without differential equations
Signal processing  : Fourier transform, filters, modulation
Mechanics          : the stability of a system is read from the SIGN of the
                     real part of the (complex) eigenvalues of its matrix
Quantum physics    : the probability amplitude is a complex number —
                     interference comes from adding these amplitudes

The common thread: what required two coupled real equations (an amplitude and a phase) becomes one complex equation.

Summary

• If Δ < 0, the solutions exist in C: z = (-b ± i√|Δ|) / 2a.
• With real coefficients, the two solutions are conjugate.
• Sum = -b/a and product = c/a still hold and serve as a check.
• Fundamental theorem of algebra: every polynomial of degree n has exactly n roots in C.
• C is algebraically closed — what R was missing.
• Applications: electricity, signal processing, mechanics, quantum physics.