Discriminant Calculator
This discriminant calculator works out b² − 4ac and what it means — and then does the thing the exam actually asks, which almost no other one will: it goes the other way and finds the missing coefficient.
Type 1, −3, 2 and you get Δ = 1, two rational roots, and the news that it factorises. Tap the small k beside a box and it solves for that box instead.
What the Discriminant Tells You
It is the piece under the square root in the quadratic formula, taken on its own:
Δ = b² − 4ac
Its whole purpose is to tell you what kind of answer exists before you go to the trouble of finding it. A square root of a negative number does not exist among the real numbers, so the sign of this one value decides everything.
The Three Cases, and a Fourth Nobody Mentions
| Δ > 0 | two real roots | x² − 4x + 1, Δ = 12 |
| Δ = 0 | one repeated root | x² − 6x + 9, Δ = 0 |
| Δ < 0 | no real roots | x² + 2x + 5, Δ = −16 |
Those three get taught everywhere, and a calculator that stops there has told you nothing you could not have worked out in ten seconds. Hiding inside the first case is a fourth question, and it is the one that actually saves time in an exam: is Δ a perfect square?
If it is, the square root comes out whole, the roots are rational, and the quadratic factorises — so you can stop reaching for the formula and just factorise it. x² − 3x + 2 has Δ = 1 = 1², and sure enough it is (x − 1)(x − 2). x² − 4x + 1 has Δ = 12, which is not a square, so it has real roots that no amount of factorising will ever find.
Where It Comes From in the Formula
Look at where Δ sits and the three cases stop being something to memorise:
x = (−b ± √Δ) ÷ 2a
Positive, and the root is a real number added and subtracted — two answers. Zero, and there is nothing to add or subtract, so the two collapse into one. Negative, and the root does not exist among the reals, so neither do the roots. The rule is just the formula, read carefully.
Finding the Value of k
Here is the question that actually appears on papers, and it runs backwards:
Find the value of k for which 2x² − 4x + k = 0 has equal roots.
Equal roots means Δ = 0, so substitute and solve — for k, not for x:
(−4)² − 4(2)(k) = 0 → 16 − 8k = 0 → k = 2
The roots of the original equation never enter into it at any point, which is exactly why it feels unfamiliar the first time. You are treating the discriminant as an equation in its own right, and k is its unknown.
The same move handles every version of the question. If the unknown is the leading coefficient — kx² + 4x + 1 = 0 with equal roots — substituting gives 16 − 4k = 0 and k = 4. If it is the constant, as above, you get one answer too. The arithmetic barely changes; what changes is which letter you are chasing.
Tap the k beside a coefficient on this page and it does precisely this, with the substitution shown.
One Answer or Two — and the k That Is Not Allowed
Where k sits decides how many answers there are.
In the a or c position, Δ is linear in k and there is one. In the b position k gets squared, so there are two: x² + kx + 4 = 0 has equal roots at k = 4 and at k = −4. Writing down only the positive one is the standard way to lose half the marks.
And there is an answer that has to be thrown away. When k is the coefficient of x², k = 0 can satisfy the arithmetic perfectly and still be wrong — it removes the x² term, and a straight line has no roots to have a nature. The algebra will not warn you; check for it every time k leads.
When the Question Is a Range, Not a Value
Change "equal roots" to "two distinct real roots" and the condition becomes Δ > 0. That is an inequality, not an equation, so there is no single k to find — the answer is every k in a range. Same for "no real roots", with Δ < 0.
It is worth noticing the shift, because a question phrased that way is not asking you to solve; it is asking you to describe. "For what values of k does this have two distinct roots?" wants an interval as its answer, and writing down a single number — even a correct boundary — answers a different question from the one on the paper.
The boundary itself is still worth finding, because it is where Δ changes sign, and that is the number the range is built around. But it belongs in the answer as an endpoint, not as the answer.
Using the Discriminant Calculator
Type a, b and c from ax² + bx + c = 0 to get Δ and what it means. To go the other way, tap the small k beside whichever coefficient is unknown, choose what the roots should be, and leave that box empty.
Fractions like 3/4 work, decimals work, and nothing is rounded. Every value of k is put back into the discriminant before it is shown to you.
Discriminant FAQ
What is the discriminant of a quadratic?
What do the three cases mean?
How do you find the value of k for equal roots?
Why are there sometimes two values of k?
Can k be zero?
What if the question asks for two distinct roots?
Sign first, then ask whether it is a square — and if the question hands you a k instead of an answer, set Δ to what it has to be and solve for that.