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Algebra

Discriminant

Tells you, before solving, whether a quadratic has two real roots, one repeated root or no real roots.

D=b2−4ac

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

See it move

−8−6−4−22468−8−6−4−22468vertex
  • Vertex (−3, 0); axis of symmetry x = −3
  • D = 0: one repeated root, x = −3

What the symbols mean

D
the discriminant (also written Δ)
a
coefficient of x2 (not 0)
b
coefficient of x
c
constant term

When to use it

Use it when a question asks about the *number* or *nature* of the roots rather than their values: "for which k does x2+kx+9=0 have equal roots?", "show the line does not meet the curve". D > 0 means two distinct real roots, D = 0 one repeated root (the graph touches the x-axis), and D < 0 no real roots. When the coefficients are whole numbers and D is a perfect square, the roots are rational and the quadratic factorises.

Worked example

How many real roots does x² + 6x + 9 = 0 have? Here a = 1, b = 6, c = 9.

  1. Start from the formula.

    D=b2−4⁢a⁢c

  2. Substitute a = 1, b = 6, c = 9 and work it out.

    D=62−4·1·9
    =36−4·9
    =36−36
    =0

  3. D = 0: the equation has exactly one (repeated) real root, and the parabola touches the x-axis at its vertex.

Answer D = 0, so there is exactly one (repeated) root

Why it works

It is the expression under the square root in the quadratic formula. A positive number has two square roots, zero has one, and a negative number has none among the real numbers.

Common mistakes

  • Forgetting that b2 is never negative: with b = −6, b2=36.
  • Reading D < 0 as "no solutions at all". There are no *real* roots; there are two complex roots.
  • Using D for a curve and a line without first forming a single quadratic equal to 0.