Quadratic Equation Solver guide
What Is the Discriminant and What Does It Tell You?
The discriminant is b² − 4ac, the part of the quadratic formula under the square root. If it is positive there are two distinct real roots; if it is zero there is one repeated root; if it is negative the roots are a complex conjugate pair and the parabola never crosses the x-axis.
What it is
The discriminant is the expression b² − 4ac — the part sitting under the square root sign in the quadratic formula. It is usually written as Δ, the Greek capital delta.
Its value tells you what kind of solutions your equation has, and it does so before you finish solving. If a question only asks how many real roots there are, the discriminant is the entire answer and you can stop there.
The three cases
| Discriminant | Roots | The parabola |
|---|---|---|
| Δ > 0 | Two distinct real roots | Crosses the x-axis twice |
| Δ = 0 | One repeated real root | Touches the x-axis once |
| Δ < 0 | Two complex conjugate roots | Never touches the x-axis |
Why the sign decides everything
The reason is simply what a square root can do.
If Δ is positive, its square root is a real number. Adding it and subtracting it gives two different answers.
If Δ is zero, the square root is zero. Adding zero and subtracting zero give the same thing, so the two branches collapse into one root.
If Δ is negative, there is no real square root — no real number squares to give a negative. The roots exist, but they involve i, the imaginary unit.
The repeated-root case
The example loaded into the calculator is x² − 6x + 9 = 0, where Δ = 36 − 36 = 0.
The single root is x = 3. It is called a repeated root or a double root, because the equation factors as (x − 3)², so 3 appears twice.
On a graph, the parabola comes down, just kisses the x-axis at x = 3, and goes back up. It never crosses.
This case is worth recognising because it often signals a perfect square in disguise, which can be a quicker route to the answer than the formula.
Perfect squares mean tidy answers
There is a second thing the discriminant tells you, beyond the count of roots.
If Δ is a perfect square — 1, 4, 9, 16, 25, 36, 49, 64 — then its square root is a whole number, and your roots will be rational. That usually also means the equation could have been factored.
If Δ is positive but not a perfect square, the roots are irrational and will involve a surd. x² − 2 = 0 has Δ = 8, giving roots of ±√2.
So a quick glance at the discriminant tells you in advance whether to expect neat answers or messy ones — useful for knowing whether you have made an arithmetic error or the question is just untidy.
Using it to find an unknown
A frequent exam question runs: "for what value of k does x² + kx + 9 = 0 have exactly one solution?"
Exactly one solution means Δ = 0. So k² − 4(1)(9) = 0, giving k² = 36, so k = 6 or k = −6.
Both work. With k = 6 the equation is (x + 3)², and with k = −6 it is (x − 3)². Losing the negative answer is the usual way to drop half the marks on this kind of question.
