Discriminant

Before you rely on it

What to check

Compute it first and read its sign.

The common error

Where people go wrong with discriminant

Trusting a calculator that returns something anyway. A negative discriminant means no real root, and a tool that hands back a number is either reporting a complex value or reporting an error.

Worked through

Discriminant in plain numbers

Take x² − 5x + 6 = 0. The discriminant b² − 4ac is 25 − 24, which is 1. Positive, so two real roots: x = 2 and x = 3. Now x² − 4x + 4 = 0 gives 16 − 16 = 0, one repeated root at x = 2. And x² + 2x + 5 = 0 gives 4 − 20 = −16, negative, so no real roots, and the two roots are complex, at −1 ± 2i. You can read the type of solutions off the discriminant before solving anything, which saves work and predicts what the graph does.

Easily confused

Discriminant and its neighbours

The discriminant is not the determinant, although the names sound alike. The determinant belongs to matrices. The discriminant belongs to polynomials and summarizes whether roots repeat. For a quadratic it is b² − 4ac, the piece under the square root in the quadratic formula. Higher-degree polynomials have discriminants too, but the formulas are longer, and a positive value means distinct real roots, zero means a repeated root and a negative value means a pair of complex roots for a quadratic with real coefficients.

In real decisions

Where discriminant turns up

Students use it to decide how many real solutions an equation has, and to check whether a parabola crosses the x-axis. Engineers use it in stability analysis of second-order systems, where the sign decides whether a response oscillates. Physicists meet it in motion problems, such as whether a projectile reaches a given height; it shows up in optimization and in checking whether a quadratic factors over the integers, when it is a perfect square.

Know the edges

Where discriminant stops being reliable

It tells you how many real roots exist, not their values. Rounding can make a tiny nonzero value look like zero, or the reverse, so treat near-zero results with care; it applies to a single polynomial equation, and its meaning changes with degree and with the number system. For coefficients that are not real numbers the sign rules do not apply in the same way. Solve the equation to get the actual roots.

Quick answers

What people ask about discriminant

What does the discriminant tell me about a parabola? If positive, the parabola crosses the x-axis twice. If zero, it just touches. If negative, it does not reach the axis.

Does a perfect-square discriminant mean the quadratic factors? For integer coefficients, yes. A perfect square gives rational roots, so the expression factors over the rationals.

How does the discriminant relate to the quadratic formula? It is the expression under the square root in the formula. If it is positive, the square root is a real number and you get two solutions. If it is zero, both solutions coincide. If it is negative, the root is imaginary and the solutions come in a complex pair.