A quadratic equation has the form $ax^2 + bx + c = 0$ with $a \ne 0$. Its roots are given by the quadratic formula, and the quantity under the square root — the discriminant $D = b^2 - 4ac$ — decides the nature of those roots.
Over the real numbers, a negative discriminant means "no solution". Once complex numbers exist, that gap closes: a negative $D$ simply produces a square root of a negative number, which we write using $i$. The result is a pair of complex roots in the form $a + ib$.
| Discriminant $D = b^2 - 4ac$ | Nature of roots |
|---|---|
| $D > 0$ | Two distinct real roots |
| $D = 0$ | One repeated real root |
| $D < 0$ | Two complex conjugate roots |
When $D < 0$, write $\sqrt{D} = \sqrt{-(4ac - b^2)} = i\sqrt{4ac - b^2}$, so the roots become:
For an equation with real coefficients, complex roots always arrive in conjugate pairs: if $p + iq$ is a root, then $p - iq$ is the other. This is why a quadratic over the reals can never have exactly one non-real root.
Relations between roots and coefficients still hold, and they hold even when the roots are complex:
These let you check an answer quickly: for conjugate roots $p \pm iq$, the sum is the real number $2p$ and the product is $p^2 + q^2$, both real, exactly as $-b/a$ and $c/a$ must be.
Deeper Insight — why complex roots are a feature, not a failure: A negative discriminant used to mark the end of the road, but it is really a signpost that the answers live one dimension over, off the real line. The reason this works so cleanly is the conjugate pairing forced by real coefficients: the imaginary parts of $p + iq$ and $p - iq$ cancel in the sum and combine into the real quantity $p^2 + q^2$ in the product, so the coefficients $-b/a$ and $c/a$ stay perfectly real even though the roots are not. This is a small instance of the Fundamental Theorem of Algebra, which promises that a degree-$n$ polynomial has exactly $n$ roots in the complex numbers, counted with multiplicity — no equation is ever truly unsolvable. The practical lesson is to stop treating $D < 0$ as an error and start reading it as an instruction: factor out the $i$, simplify the surd, and report the conjugate pair.