the complex-conjugate-roots case

When does a system oscillate rather than just decay? When the characteristic equation has no real roots at all, but a pair of complex numbers instead. This is the case of every spring that wobbles, every pendulum that swings, every tuned circuit that rings: the mathematics of vibration is hiding inside an imaginary number.

If a r^2 + b r + c = 0 has discriminant b^2 - 4ac < 0, its roots are a complex-conjugate pair r = alpha ± beta i, with alpha = -b/(2a) the real part and beta = sqrt(4ac - b^2)/(2a) the imaginary part. The formal solutions e^((alpha + beta i)x) and e^((alpha - beta i)x) involve i, but we want real-valued solutions. Euler's formula e^(i theta) = cos(theta) + i sin(theta) rescues us: writing e^((alpha ± beta i)x) = e^(alpha x)(cos(beta x) ± i sin(beta x)) and taking real and imaginary parts gives two real, independent solutions, e^(alpha x) cos(beta x) and e^(alpha x) sin(beta x). The general solution is y = e^(alpha x)(c1 cos(beta x) + c2 sin(beta x)).

Now the two parts of the root tell a clean physical story. The real part alpha controls the envelope: alpha < 0 means the oscillation decays (a damped vibration settling down), alpha = 0 means it persists forever at constant amplitude (pure simple harmonic motion), alpha > 0 means it grows. The imaginary part beta is the angular frequency of the wiggle. So one complex root encodes both how fast the system shakes and how quickly that shaking fades — the entire qualitative behaviour of an oscillator in two real numbers.