The discriminant is the part of the quadratic formula that sits under the square root sign: b² − 4ac. Its value tells you how many real roots a quadratic equation has without you solving the equation at all. Positive means two distinct roots, zero means one repeated root, and negative means no real roots.
Where does the discriminant come from?
Every quadratic equation can be written in the form ax² + bx + c = 0, where a is not zero. The quadratic formula solves it:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
The whole behaviour of that formula depends on what happens inside the square root. That expression, b² − 4ac, is called the discriminant, because it discriminates between the three possible cases.
- If b² − 4ac is positive, the square root gives a real number, and the ± produces two different answers. Two distinct real roots.
- If b² − 4ac is zero, the square root is zero, so adding and subtracting it gives the same answer twice. One repeated root (sometimes called equal roots).
- If b² − 4ac is negative, you would need the square root of a negative number, which does not exist in the real numbers you use at GCSE. No real roots.
How do you calculate the discriminant?
Follow the same four steps every time.
- Rearrange the equation into the form ax² + bx + c = 0. Everything must be on one side, with zero on the other. This is the step students most often skip, and it is where most marks are lost.
- Write down a, b and c, including their signs. A minus sign in front of a term belongs to the coefficient.
- Substitute into b² − 4ac. Put brackets around any negative value before squaring it.
- Compare the result with zero and state the conclusion in words.
Worked examples: the three cases
Example 1 — two distinct roots. How many real roots does x² − 5x + 6 = 0 have?
Here a = 1, b = −5, c = 6.
b² − 4ac = (−5)² − 4 × 1 × 6 = 25 − 24 = 1
The discriminant is positive, so the equation has two distinct real roots. (Factorising confirms it: (x − 2)(x − 3) = 0, giving x = 2 and x = 3.)
Example 2 — one repeated root. How many real roots does x² − 6x + 9 = 0 have?
Here a = 1, b = −6, c = 9.
b² − 4ac = (−6)² − 4 × 1 × 9 = 36 − 36 = 0
The discriminant is zero, so there is one repeated root. Indeed x² − 6x + 9 = (x − 3)², so x = 3 is the only solution.
Example 3 — no real roots. How many real roots does 2x² + 3x + 5 = 0 have?
Here a = 2, b = 3, c = 5.
b² − 4ac = 3² − 4 × 2 × 5 = 9 − 40 = −31
The discriminant is negative, so the equation has no real roots.
What does the discriminant tell you about the graph?
The roots of ax² + bx + c = 0 are exactly the x-coordinates where the curve y = ax² + bx + c crosses the x-axis. So the discriminant describes the picture:
| Discriminant | Roots of the equation | The graph and the x-axis |
|---|---|---|
| b² − 4ac > 0 | Two distinct real roots | Crosses the x-axis twice |
| b² − 4ac = 0 | One repeated root | Touches the x-axis once (the turning point sits on the axis) |
| b² − 4ac < 0 | No real roots | Never meets the x-axis |
This is why exam questions sometimes ask you to "show that the curve does not cross the x-axis" — they are asking you to show the discriminant is negative.
How do you answer "find the value of k" questions?
Higher-tier papers usually hide the discriminant inside a question about an unknown coefficient.
Question: The equation x² + kx + 9 = 0 has equal roots. Find the possible values of k.
"Equal roots" means the discriminant is zero, so:
k² − 4 × 1 × 9 = 0 k² − 36 = 0 k² = 36 k = 6 or k = −6
Notice that both values are needed — dropping the negative solution is the classic lost mark here.
Question: The equation x² + (k + 2)x + 9 = 0 has two distinct real roots. Find the range of values of k.
Two distinct real roots means the discriminant is greater than zero:
(k + 2)² − 36 > 0 (k + 2)² > 36 k + 2 > 6 or k + 2 < −6 k > 4 or k < −8
Because the inequality involves a square, it produces two separate regions, not one interval. Sketching y = (k + 2)² − 36 makes this obvious: the expression is positive outside the two roots.
How is the discriminant used in tangent questions?
If a straight line is a tangent to a curve, it touches the curve exactly once — which means the equation you get by setting them equal has one repeated root, so its discriminant is zero.
Question: The line y = x + k is a tangent to the curve y = x². Find k.
Set them equal: x² = x + k, so x² − x − k = 0.
Here a = 1, b = −1, c = −k. For a tangent, the discriminant is zero:
(−1)² − 4 × 1 × (−k) = 0 1 + 4k = 0 k = −¼
The same method answers "show that the line and curve never meet" (discriminant negative) and "show that the line crosses the curve twice" (discriminant positive).
Common mistakes to avoid
- Not rearranging to = 0 first. For x² = 4x − 7 you must write x² − 4x + 7 = 0 before reading off a, b and c.
- Losing a minus sign when squaring. (−5)² is 25, not −25. Always use brackets.
- Forgetting the sign of c. In x² − 3x − 10 = 0, c is −10, so −4ac = −4 × 1 × (−10) = +40.
- Saying "no solutions" instead of "no real roots". The precise wording is what examiners mark.
- Giving only the positive value of k in equal-roots questions.
Frequently asked questions
Is the discriminant on both foundation and higher tier?
The discriminant is a higher-tier topic. Foundation-tier students solve quadratics by factorising and use the quadratic formula where it is given, but they are not asked to reason about the number of roots from b² − 4ac. If you are entered for higher tier, treat it as a topic you must know, since it appears both on its own and hidden inside tangent and "find k" questions.
What does it mean if the discriminant is a perfect square?
If b² − 4ac is a positive perfect square, such as 1, 4, 9 or 25, the square root is a whole number and the two roots are rational — which tells you the quadratic would have factorised. If it is positive but not a perfect square, the roots are still real but irrational, and you would leave them as surds or round them as the question instructs.
Does a negative discriminant mean the equation has no solutions at all?
It means there are no real solutions, which is the correct answer at GCSE. In further mathematics you meet complex numbers, where a negative discriminant gives two complex solutions. At GCSE you should write "no real roots" or "no real solutions", and if you are sketching the curve, show it sitting entirely above or entirely below the x-axis.
Do I need to memorise b² − 4ac, or is it on the formula sheet?
The GCSE formula sheet gives the quadratic formula, and the discriminant is the expression under the root sign within it, so you can always recover it. What is not given is the meaning — that positive, zero and negative correspond to two roots, one repeated root and no real roots. Memorise the three cases and their graph pictures.
For Socratic GCSE algebra practice with Professor Pi — who will lead you to the discriminant rather than hand you the answer — visit aitutors.me.