Quadratic Regression Calculator
Enter paired X and Y data to fit a least-squares quadratic y = cx² + bx + a. Get R², all three coefficients, the parabola equation, and a prediction at any new X.
Quadratic equation: y = 1x² + 2x − 1
How does this calculator work?
Quadratic regression fits y = cx² + bx + a by solving a 3×3 normal-equation system from paired X, Y data. Needs ≥ 3 points. R² = 1 − SSresiduals/SStotal; values near 1 mean a tight parabolic fit. c > 0 opens upward, c < 0 downward. Vertex at x = −b/(2c). Extrapolation beyond observed X is unreliable.
Formula
How this is calculated
Quadratic regression finds the three coefficients a, b, c of the parabola y = a + bx + cx² that minimise the total squared residual error Σ(yᵢ − ŷᵢ)². The optimal values satisfy a 3×3 linear system called the normal equations, built from the power sums Σxᵏ (k = 0..4) and the cross-products Σxᵏy (k = 0..2). This calculator assembles that system and solves it with Gaussian elimination with partial pivoting for numerical stability.
At least three data points are required because the parabola has three free parameters. With exactly three the curve passes through all of them (R² = 1 by construction); more data than parameters is needed for a meaningful statistical fit, so at least five or six points are recommended. R² = 1 − SSresiduals/SStotal: values near 1 indicate a tight parabolic fit; lower values mean the relationship is not well described by a quadratic.
The sign of c determines concavity: c > 0 gives an upward-opening parabola (minimum), c < 0 a downward one (maximum). The vertex is at x = −b/(2c). Extrapolation beyond the observed X range is unreliable — quadratics grow without bound for large |x|. For ill-conditioned data (very wide X ranges), centre the X values by subtracting their mean before fitting.
Frequently asked questions
A quadratic polynomial has 3 unknown coefficients (a, b, c). Three points exactly determine a unique parabola through them. More points give a least-squares estimate of the true underlying relationship and yield a meaningful R² statistic.
R² = 1 − SSresiduals/SStotal measures the fraction of variance in Y explained by the fitted parabola. R² near 1 means an excellent quadratic fit; near 0 means the parabola explains little variation. Higher-degree polynomials always raise R² — use adjusted R² or cross-validation to compare models.
Use quadratic when a scatter plot shows a curved (U-shaped or inverted-U) pattern rather than a straight line. If linear regression leaves systematic curvature in the residuals, quadratic regression is a natural next step. If the relationship is still curved after fitting a quadratic, try cubic or a different model.
Also known as
TG we-Calculate Editorial Team. (2026). Quadratic Regression Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/quadratic-regression-calculator
TG we-Calculate Editorial Team. "Quadratic Regression Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/quadratic-regression-calculator.
TG we-Calculate Editorial Team, "Quadratic Regression Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/quadratic-regression-calculator
@misc{wecalculate_quadratic_regression_calculator, title = {Quadratic Regression Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/quadratic-regression-calculator}}, year = {2026}, note = {TG we-Calculate} }
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