The discriminant of ax² + bx + c = 0 is b² − 4ac. If it is positive there are two distinct real roots, if it is zero there is one repeated root, and if it is negative there are no real roots. Geometrically it counts how many times a curve meets the x-axis or a line, so conditions such as “equal roots”, “two distinct points” or “a tangent” all become statements about the discriminant.
The discriminant
- Discriminant: b² − 4ac
- b² − 4ac > 0: two distinct real roots
- b² − 4ac = 0: one repeated (equal) root
- b² − 4ac < 0: no real roots
- A line is a tangent to a curve when the resulting quadratic has b² − 4ac = 0
What does the discriminant tell you?
The discriminant is the part under the square root in the quadratic formula. If it is positive, the square root gives two different values and there are two distinct roots. If it is zero there is just one repeated root. If it is negative there are no real roots, because a negative number has no real square root.
Rearrange the equation to ax² + bx + c = 0 before reading off a, b and c; using the wrong c is a common slip.
Conditions on an unknown
Exam questions turn worded conditions into discriminant statements. Equal or repeated roots means b² − 4ac = 0. Two distinct real roots means b² − 4ac > 0. No real roots means b² − 4ac < 0. Real roots, which allows equal ones, means b² − 4ac ≥ 0. If the unknown appears in more than one coefficient, the discriminant becomes a quadratic in the unknown, which you then solve, often as an inequality.
The discriminant and the graph
The discriminant counts the points where y = ax² + bx + c meets the x-axis: twice if positive, once (touching) if zero, never if negative. To show that a curve lies entirely above the x-axis, you need two facts together: it opens upwards (a > 0) and it never touches the axis (b² − 4ac < 0).
Tangents and lines meeting curves
To find how many times a line meets a curve, substitute the line into the curve to form a quadratic, then use its discriminant. A discriminant of zero is exactly the condition for the line to be a tangent. If the line's gradient or intercept is unknown, setting the discriminant to zero gives an equation for it.
Proving results for all values
The hardest marks ask you to show something holds for all values of k. Often the discriminant simplifies to a perfect square, such as (k − 1)², which is never negative, so the roots are real for every k. Other questions need the full set of values satisfying a quadratic inequality in k.
Worked examples
Example 1. Determine the nature of the roots of 2x² − 5x + 1 = 0.
- a = 2, b = −5, c = 1.
- b² − 4ac = 25 − 8 = 17.
Answer: 17 > 0, so there are two distinct real roots.
Marks: M1 for substituting, A1 for 17 and the conclusion.
Example 2. The equation x² + kx + 9 = 0 has equal roots. Find k.
- Equal roots: b² − 4ac = 0.
- k² − 36 = 0.
Answer: k = ±6.
Marks: M1 for k² − 36 = 0, A1 for k = ±6.
Example 3. Show that y = x² + 2x + 5 lies entirely above the x-axis.
- a = 1 > 0, so the curve opens upwards.
- b² − 4ac = 4 − 20 = −16 < 0, so it never meets the x-axis.
Answer: The curve opens upwards and has no real roots, so it lies entirely above the x-axis.
Marks: M1 for the discriminant, A1 for the full conclusion.
Example 4. Show that y = 2x − 1 is a tangent to y = x² − 2x + 3.
- Substitute: x² − 2x + 3 = 2x − 1, so x² − 4x + 4 = 0.
- b² − 4ac = 16 − 16 = 0.
Answer: The discriminant is 0, so the line touches the curve once, at x = 2: it is a tangent.
Marks: M1 for the quadratic, A1 for the discriminant and conclusion.
Example 5. Show that x² + (k + 1)x + k = 0 has real roots for all real values of k.
- b² − 4ac = (k + 1)² − 4k.
- = k² + 2k + 1 − 4k = k² − 2k + 1 = (k − 1)².
Answer: (k − 1)² ≥ 0 for all k, so the discriminant is never negative and the roots are always real.
Marks: M1 for the discriminant, A1 for (k − 1)² ≥ 0 and the conclusion.
Common mistakes
- Reading c before rearranging. Write the equation as ax² + bx + c = 0 first.
- Using > 0 for “real roots”. Real roots allows equal roots, so use b² − 4ac ≥ 0.
- Forgetting the a > 0 condition. To be above the x-axis the curve must also open upwards.
- Solving a quadratic inequality wrongly. Sketch the quadratic in k to see whether the solution is between or outside the roots.
- Substituting the wrong way round. Substitute the line into the curve to get one quadratic in x.
How the discriminant is examined
The discriminant sits in the quadratics chapter of Pure 1 (WMA11) in Edexcel International A-Level, and in first-year Pure for Edexcel A-Level (9MA0). It is a favourite source of multi-step marks:
- find the values of k for equal, distinct or no real roots
- show that a quadratic inequality in k holds, then solve it
- show a line is a tangent, or find the tangents from a point
- prove an equation has real roots for all values of a constant
Multi-step discriminant questions with full mark schemes are in the Pure 1 quadratics exam questions. Practise further with the Pure 1 quadratics worksheet or the A-Level algebra revision engine.
Solving the inequalities needs quadratic inequalities. The same tangent condition appears in tangents to circles.
Frequently asked questions
What is the discriminant?
b² − 4ac, the expression under the square root in the quadratic formula.
What does a discriminant of zero mean?
The quadratic has one repeated root, and its graph just touches the x-axis. For a line and a curve, it means the line is a tangent.
What is the difference between “real roots” and “distinct real roots”?
Real roots means b² − 4ac ≥ 0, which allows equal roots. Distinct real roots means b² − 4ac > 0.