Complex Roots of Quadratics

With complex numbers available, every quadratic has exactly two roots. The negative discriminant is no longer a dead end.

The method

Use the formula as usual. When the discriminant is negative, write its root using i.

They come in pairs

With real coefficients, complex roots are always conjugates. The ± in the formula guarantees it.

What the graph shows

Complex roots mean the parabola never crosses the x-axis. The roots are real solutions to nothing you can see.

The vertex is still there

A parabola with complex roots still has a perfectly ordinary vertex and y-intercept.

Checking

Substituting a complex root back into the quadratic should give zero. The i² terms are what make it work out.

Every quadratic has two roots

Allowing complex numbers, every quadratic has exactly two roots, counted with multiplicity. The negative discriminant case is no longer "no solution" but two complex ones.

They come in conjugate pairs

A quadratic with real coefficients has complex roots that are conjugates of each other. The ± in the formula is what produces the pair, so they always arrive together.

Complex roots mean no x-intercepts

A parabola with complex roots floats entirely above or below the axis. The roots exist algebraically and are invisible on the real graph, which is a useful thing to be able to state.

The pattern generalises

Every polynomial of degree n has exactly n complex roots, counted with multiplicity. That is the fundamental theorem of algebra, and quadratics are its simplest interesting case.