Quadratic Functions & Complex Numbers

The Discriminant

Using the discriminant to determine the number and type of solutions to a quadratic.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Operations with Complex Numbers

Completing the Discriminant's Story

In Algebra I, a negative discriminant meant 'no real solutions,' and the story stopped there. Before reading on, think about what actually happens, algebraically, when the quadratic formula's discriminant is negative — now that you have a tool that handles negative square roots.

When b² − 4ac is negative, √(b² − 4ac) simplifies to a multiple of i, and the quadratic formula produces two complex solutions instead of failing outright. The discriminant now fully classifies every quadratic's solution type, with no more exceptions: positive gives two distinct real solutions, zero gives one repeated real solution, and negative gives two complex solutions.

Those two complex solutions are always a conjugate pair — and this follows directly from the formula's structure. x = (−b ± √(b² − 4ac)) / (2a): once the discriminant is negative, the ± in front of the (now imaginary) square root produces two solutions differing only in the sign of their imaginary part — exactly the definition of complex conjugates.

Worked Example — Classifying Solutions by the Discriminant

Classify the solutions of x² − 6x + 13 = 0 without fully solving. Compute the discriminant: b² − 4ac = 36 − 52 = −16. Since it's negative, the equation has two complex conjugate solutions.

Worked Example — Verifying a Conjugate Pair

Solve x² − 6x + 13 = 0 fully. x = (6 ± √(−16))/2 = (6 ± 4i)/2 = 3 ± 2i. The two solutions, 3 + 2i and 3 − 2i, are indeed complex conjugates, as predicted by the negative discriminant.

b^{2} - 4ac < 0 \;\Longrightarrow\; x = \dfrac{-b}{2a} \pm \dfrac{\sqrt{4ac-b^2}}{2a}i

Tip

Checking the discriminant's sign before fully solving predicts exactly what kind of answer to expect — a negative discriminant means the final answer should come out as a conjugate pair, a useful check against an arithmetic slip.

Common Mistakes

Stopping at 'no real solutions' for a negative discriminant, without recognizing that complex solutions still exist and can be found.
A negative discriminant means no real solutions specifically, not no solutions at all — the quadratic formula still produces two valid complex solutions.
Reporting only one of the two complex solutions, forgetting the ± produces a conjugate pair.
Just like the real-solution case, the ± in the quadratic formula always produces two solutions — for a negative discriminant, report both halves of the conjugate pair.

Key Takeaways

The discriminant now fully classifies a quadratic's solutions: positive (two real), zero (one repeated real), or negative (two complex).
A negative discriminant's two complex solutions are always a conjugate pair, since the ± only affects the imaginary part.
The discriminant's sign predicts the type of answer to expect before fully solving.

Summary

The discriminant, together with complex numbers, gives every quadratic equation a complete solution — no exceptions left. The next lesson works through solving quadratics with complex roots in full, and connects the result back to the parabola's graph.

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