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MathematicsGCSE Higher & A Level

What is discriminant?

The discriminant, b² − 4ac, tells you how many real roots a quadratic equation has without solving it.

Discriminant explained

A positive discriminant means two distinct real roots, zero means one repeated root, and a negative value means no real roots — the curve never crosses the x-axis.
Exam questions often reverse this: they give a condition on the roots and ask you to form and solve an inequality in an unknown coefficient.

Key formula

Δ=b2−4ac\Delta = b^{2} - 4ac

Worked example

x2+kx+9=0x^{2} + kx + 9 = 0 has equal roots when k2−36=0k^{2} - 36 = 0, so k=±6k = \pm 6.

Examiner tip

State the condition in words — 'equal roots so the discriminant is zero' — before doing algebra. That sentence is usually worth a mark.

Discriminant: common questions

What does a negative discriminant mean graphically?
The parabola lies entirely above or entirely below the x-axis, so there are no real solutions — only complex ones in Further Maths.
Does the discriminant work for cubics?
There is a cubic discriminant, but it is well beyond A Level. At this level, apply it only to quadratics.

Related terms

Quadratic formula

The quadratic formula solves any equation of the form ax² + bx + c = 0, including those that will not factorise.

Completing the square

Completing the square rewrites a quadratic in the form a(x + p)² + q, revealing the turning point and allowing exact solutions.

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