Quadratic Equation Calculator
Enter coefficients for ax2 + bx + c = 0. This tool returns roots, discriminant, vertex, axis of symmetry, and basic parabola behavior.
What is a quadratic equation?
A quadratic equation is any equation that can be written in the form ax2 + bx + c = 0 where a ≠ 0. The graph of this equation is a parabola. Depending on the values of a, b, and c, the parabola can open upward or downward, cross the x-axis in two places, touch it once, or never cross it at all.
What this calculator gives you
- Discriminant (Δ): Δ = b2 - 4ac
- Roots: real or complex solutions for x
- Axis of symmetry: x = -b / (2a)
- Vertex: the turning point of the parabola
- Direction: whether it opens up (a > 0) or down (a < 0)
- Sum/product of roots: quick checks from Vieta’s formulas
How to interpret the discriminant
1) If Δ > 0
You get two distinct real roots. The parabola crosses the x-axis at two different points.
2) If Δ = 0
You get one repeated real root. The parabola just touches the x-axis at its vertex.
3) If Δ < 0
You get two complex conjugate roots. The parabola does not cross the x-axis in the real plane.
Example walkthrough
Try the common example x2 - 5x + 6 = 0:
- a = 1, b = -5, c = 6
- Δ = (-5)2 - 4(1)(6) = 25 - 24 = 1
- Since Δ > 0, there are two real roots
- Roots are x = 2 and x = 3
This matches factoring: (x - 2)(x - 3) = 0.
When this is useful
Quadratics appear constantly in science, engineering, and finance. A few examples:
- Physics: projectile motion and time-to-impact
- Optimization: maximum profit/minimum cost models
- Geometry: area and dimension constraints
- Data modeling: simple curve fitting and trend analysis
Common mistakes to avoid
- Setting a = 0 (that would make it linear, not quadratic)
- Sign errors inside the discriminant, especially with negative b or c
- Forgetting that complex roots come in conjugate pairs
- Rounding too early and introducing avoidable error
Tip: If your roots seem odd, compare the calculator’s sum and product of roots to -b/a and c/a. It’s a fast consistency check.
Final thought
A good quadratic calculator should do more than return two numbers. It should help you understand the equation’s structure: roots, shape, and turning point. Use this tool as both a solver and a learning aid.