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What Is the Discriminant?

Q: What does a discriminant of zero mean?

The quadratic has one repeated root rather than two, and its graph touches the x-axis at a single point instead of crossing it. Questions often phrase this as equal roots, or say the line is a tangent to the curve, which amounts to the same condition: b² − 4ac = 0.

Q: Can the discriminant give you the roots themselves?

No, only how many there are. Finding the roots means putting the discriminant back inside the quadratic formula, or factorising, or completing the square. The discriminant is a counting tool, which is exactly why it is useful — it answers the how-many question without the work of answering the what-are-they question.

Q: What does a negative discriminant mean?

That the equation has no real solutions. The parabola misses the x-axis completely, sitting entirely above it or entirely below it. At GCSE and IGCSE that is the end of the story. Beyond school the equation does have solutions, but they are complex numbers rather than points on the number line.

Q: Why is it called the discriminant?

Because it discriminates between the cases — the word comes from the Latin for distinguishing one thing from another. It is a single number whose job is to sort every quadratic into one of three groups, and it has no other meaning attached to it.

The sign of b² − 4ac settles whether a quadratic has two real roots, one, or none — computed for three equations, with the graph each case produces.

Discriminant

The part of the quadratic formula under the square root, b² − 4ac, whose sign tells you whether a quadratic has two real roots, one, or none.

أين يقابله الطلاب: Grade 10 mathematics, on GCSE Higher tier and IGCSE Extended papers, usually within a lesson or two of the quadratic formula itself.

الإجابة باختصار
The discriminant is b² − 4ac, the expression under the square root in the quadratic formula. Its sign settles how many real roots a quadratic has: positive gives two, zero gives one repeated root, negative gives none. You can work it out without solving the equation at all.

مثال

x² − 5x + 6 → 1 x² − 6x + 9 → 0 x² + x + 3 → −11

For x² − 5x + 6 = 0, a = 1, b = −5 and c = 6, so b² − 4ac = 25 − 24 = 1. Positive, so two roots, and the parabola cuts the x-axis twice — at 2 and at 3. For x² − 6x + 9 it is 36 − 36 = 0: one repeated root at x = 3, where the curve comes down, touches the axis and turns. For x² + x + 3 it is 1 − 12 = −11, and no real number squares to give a negative, so that curve sits entirely above the axis and never meets it.

Three quadratics, three shapes

Computing it is substitution and nothing else, but one slip accounts for most wrong answers: b² when b is negative. In x² − 5x + 6, b is −5, so b² is 25, not −25. Bracket the value before squaring it and the error disappears.

Once you have the number, the sign tells you the shape of the graph before you have plotted a single point. That is the real use of it — the discriminant is a way of seeing the picture cheaply.

b² − 4ac > 0 — two different real roots, the curve crosses the x-axis twice
b² − 4ac = 0 — one repeated root, the curve touches the x-axis and turns
b² − 4ac < 0 — no real roots, the curve never meets the x-axis

Why the sign is doing the work

In the formula x = (−b ± √(b² − 4ac)) / 2a, the discriminant is the number you take the square root of. A positive number has two square roots, one either side of zero, so the ± produces two different answers. Zero has one square root, so the ± adds and subtracts nothing and both branches give the same answer. A negative number has no real square root, so the formula produces nothing real to report.

There is a further reading worth having. When the discriminant is a positive square number — 1, 4, 9, 16 — the root comes out whole, the answers are fractions or integers, and the quadratic factorises with integer brackets. The 1 above is why x² − 5x + 6 splits neatly into (x − 2)(x − 3). When it is positive but not a square number, the roots are surds and factorising will not work.

The question it is really set for

Papers rarely ask for the discriminant by name. They ask you to show that an equation has no real solutions, or they hide a letter in the coefficients and ask which values of it give equal roots — and that second version is the one worth rehearsing, because it turns a one-line check into an equation of its own.

For x² + kx + 9 = 0 to have equal roots, the discriminant must be zero: k² − 36 = 0, so k = 6 or k = −6. Students who solve k² = 36 and write only k = 6 have done the harder half of the work and lost the mark on the easier half.

أسئلة شائعة

What does a discriminant of zero mean?