The short answer
The quadratic formula solves any equation of the form ax² + bx + c = 0: x = (−b ± √(b²−4ac)) ÷ (2a). The ± means there are two solutions, found by adding and then subtracting the square root term. The discriminant, b²−4ac, previews which kind of solutions to expect before you even finish the calculation.
Key takeaways
- The discriminant (b²−4ac) previews the answer type: positive means two real roots, zero means one repeated real root, negative means two complex roots.
- The vertex, at x = −b/(2a), is the parabola's minimum point if a > 0 or its maximum point if a < 0.
- Factoring is faster than the formula when a quadratic factors neatly over integers, but the formula always works, even when factoring doesn't.
- A negative discriminant doesn't mean "no solution" — it means no real solution; the two complex solutions still exist and follow the same formula.
The quadratic formula and the discriminant
x = (−b ± √(b² − 4ac)) / (2a)
| Discriminant | Solution type | Example |
|---|---|---|
| b²−4ac > 0 | Two real solutions | x²−5x+6=0 |
| b²−4ac = 0 | One repeated real solution | x²−4x+4=0 |
| b²−4ac < 0 | Two complex solutions | x²+x+1=0 |
Worked example: solving x² − 5x + 6 = 0
a=1, b=−5, c=6
Discriminant = (−5)² − 4(1)(6) = 25 − 24 = 1
x = (5 ± 1) / 2 → x = 3 or x = 2
This matches factoring the same equation directly: x² − 5x + 6 = (x − 2)(x − 3), which is zero exactly when x = 2 or x = 3 — the quadratic formula and factoring agree, as they always will.
Vertex and axis of symmetry
x = −b/(2a) = −(−5)/(2×1) = 2.5
y = (2.5)² − 5(2.5) + 6 = 6.25 − 12.5 + 6 = −0.25
Vertex = (2.5, −0.25)
Since a = 1 is positive, this parabola opens upward, so the vertex (2.5, −0.25) is the lowest point on the curve — and it sits exactly halfway between the two roots, 2 and 3, on the axis of symmetry x = 2.5.
Common mistakes to avoid
- Forgetting the ± and reporting only one root — every quadratic with a positive discriminant has two distinct solutions.
- Sign errors when b is negative — a negative b makes −b positive, a frequent source of arithmetic mistakes.
- Treating a negative discriminant as "no solution" instead of "no real solution" — the two complex solutions still exist mathematically.
- Stopping at x = −b/(2a) for the vertex and forgetting to plug it back into the original equation to get the y-coordinate too.
Related calculators
- Root Calculator — double-check the square root step inside the formula.
- Exponent Calculator — verify the squared terms in the discriminant.
- Matrix Calculator — solve systems of equations beyond a single quadratic.
- Scientific Calculator — handle general algebraic and trigonometric calculations.