The formula
Δ = b² − 4ac
Why it works
The discriminant is the part of the quadratic formula under the square root. Since a square root of a negative number is not real, that single expression decides everything: positive gives two answers, zero gives one, negative gives none.
How to solve it by hand
- Square b
- Multiply 4 by a by c
- Subtract the second from the first
- Read the sign: positive, zero or negative
What is worth knowing
The geometry makes it intuitive. A quadratic graphs as a parabola, and its real roots are exactly where that parabola crosses the x-axis. Two crossings means two roots; touching the axis at its vertex means one repeated root; floating entirely above or below means none. That is also why a zero discriminant is called a repeated root rather than a single one: both solutions are there, they have just landed on top of each other. In practice the discriminant is most useful when you only need to know whether a solution exists — in optimisation, in physics, or when checking whether two curves can intersect at all.