Quadratics with Complex Roots

In Algebra 1, a negative discriminant meant "no solution." With complex numbers, it means two complex solutions instead.

For x² + 2x + 5 = 0: b² − 4ac = 4 − 20 = −16.

Use the quadratic formula as usual, then extract i:
x = (−2 ± √(−16))/2 = (−2 ± 4i)/2 = −1 ± 2i.

Complex roots always arrive in conjugate pairs — if −1 + 2i is a solution, so is −1 − 2i. You never get just one.

On the graph, a negative discriminant means the parabola never touches the x-axis. The solutions are real mathematics, they simply are not visible as x-intercepts.

For real coefficients, the discriminant Δ = b² − 4ac classifies the roots: Δ > 0 two distinct real, Δ = 0 one repeated real, Δ < 0 two non-real conjugates p ± qi.

The conjugate pairing is guaranteed by the Complex Conjugate Root Theorem: if a polynomial has real coefficients and z is a root, then z̄ is also a root. This is why non-real roots always occur in pairs and why any real polynomial of odd degree must have at least one real root.

Vieta's formulas continue to hold over ℂ: the sum of roots is −b/a and the product c/a. For −1 ± 2i the sum is −2 and the product is (−1)² + 2² = 5, matching x² + 2x + 5 — a fast verification that needs no re-substitution.

Graphically, Δ < 0 means the parabola misses the x-axis entirely. The roots are still meaningful: in engineering they describe oscillating rather than decaying behaviour, which is why complex roots are read as a physical result and not as failure.