About Quadratic Formula Calculator
This Quadratic Formula Calculator solves equations of the form ax² + bx + c = 0, providing both real and complex roots, the discriminant, the nature of the roots, and the sum and product of the roots. It also plots the parabola on an interactive chart so you can visualize the function and its x-intercepts. The discriminant (Δ = b² − 4ac) tells you everything about the roots: positive means two distinct real roots, zero means one repeated real root, and negative means two complex conjugate roots. Quadratic equations appear throughout algebra, physics, engineering, and economics — from calculating projectile trajectories to optimizing profit functions. The sum and product of roots relationships (given by −b/a and c/a) provide a quick sanity check. The chart uses Chart.js to render the curve, showing the vertex and the roots as x-intercepts when they are real. This calculator is perfect for high school and college students checking their work, for teachers preparing examples, and for professionals who occasionally need to solve quadratic models quickly.
How to Use This Calculator
Let's solve the equation x² - 5x + 6 = 0. Enter a = 1, b = -5, and c = 6, then click Calculate. The discriminant is 1 (positive), so there are two distinct real roots. The calculator shows x₁ = 3 and x₂ = 2. The sum of roots is 5 (which equals -b/a = 5) and the product is 6 (which equals c/a = 6). The chart plots the parabola crossing the x-axis at x = 2 and x = 3. Now try a case with complex roots: x² + 2x + 5 = 0 (a = 1, b = 2, c = 5). The discriminant is -16, so the roots are x = -1 ± 2i — shown as complex numbers in the result. The chart shows the parabola hovering above the x-axis with no intercepts.
How to Interpret Your Results
The discriminant is your first clue. If Δ > 0, like when a = 1, b = -7, c = 10 yields Δ = 9, the parabola crosses the x-axis at two points (x = 2 and x = 5). If Δ = 0, the parabola just touches the x-axis at one point — the vertex sits right on the axis. For example, x² - 6x + 9 = 0 has Δ = 0 and the only root is x = 3. If Δ < 0, like x² + 4x + 7 = 0 with Δ = -12, the parabola never touches the x-axis, and the roots are complex. The sum and product also tell you something: they equal -b/a and c/a respectively, which is useful for quickly checking if you factored correctly.