complex root

A complex root is a solution of a polynomial equation that lives among the complex numbers, and the term usually highlights the non-real ones — solutions with a genuine imaginary part. When a parabola never crosses the x-axis, for example, its equation still has solutions; they are simply hiding off the real number line as complex roots.

Strictly speaking every real number is also a complex number, so all roots of a polynomial are complex roots in the broad sense. In everyday usage, though, calling something a complex root signals that it is not real. For a real-coefficient polynomial these non-real roots always appear in conjugate pairs: if a + bi is a root, so is a - bi.

For a quadratic, the discriminant b^2 - 4ac is the tell-tale sign: when it is negative, the two roots are a conjugate pair of complex numbers rather than real numbers. The quadratic formula still applies — you just take the square root of a negative number and write it using i.