Quadratic Regression Calculator (Least Squares)

Fit a quadratic regression y=a+bx+cx² to your (x, y) data points by least squares. Get the coefficients, R², and the parabola vertex, with a scatter plot and fitted curve.

Input

Enter your (x, y) data points to fit the quadratic regression y = a + bx + cx² by least squares.

Data points (x, y)

One point per line. Separate x and y with a comma or a space (e.g. 1, 3). At least 3 points are required.

Result

Quadratic regression equation

y = 4.6 − 2.9571x + 1.5x²

Constant a

4.6

Linear term b

-2.957143

Quadratic term c

1.5

R squared

0.9997

Data points

Vertex

(0.9857, 3.1426)

The quadratic coefficient is positive, so the parabola opens upward and the vertex is a minimum.

Scatter plot and fitted curve

Data points and predictions

x	Observed y	Predicted	Residual
1	3	3.1429	-0.1429
2	5	4.6857	0.3143
3	9	9.2286	-0.2286
4	17	16.7714	0.2286
5	27	27.3143	-0.3143
6	41	40.8571	0.1429

How it works

Quadratic regression fits the second-degree model y=a+bx+cx² to your (x, y) data points. The coefficients a, b, and c are chosen by least squares so that the sum of the squared residuals (the gaps between observed and predicted values) is as small as possible.
The tool builds the normal equations, a system of three linear equations, and solves them with Gaussian elimination using partial pivoting to obtain a, b, and c. At least three data points are required.
The coefficient of determination R² measures how well the model explains the variation in the data, on a scale from 0 to 1. Values closer to 1 indicate a better fit.
The parabola vertex is found by solving dy/dx=b+2cx=0, giving x=−b÷(2c). When the quadratic coefficient c is positive the curve opens upward and has a minimum; when c is negative it opens downward and has a maximum.
Enter one point per line with x and y separated by a comma or a space. The table also lists the predicted value and residual for each point.

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