Quadratic Form Calculator

This quadratic form calculator evaluates a two-variable quadratic form, Q(x, y) = a x squared plus b xy plus c y squared, at any point you choose, and tells you whether the form is positive definite, negative definite, indefinite or degenerate. You enter the three coefficients, a, b and c, along with the x and y values of the point you want to test. The calculator returns the value of Q at that point, the definiteness classification, and the determinant of the symmetric matrix behind the form, calculated as ac minus b squared over four. Definiteness matters because it describes the shape of the underlying surface: a positive definite form has a genuine minimum, a negative definite form has a genuine maximum, and an indefinite form has a saddle point, distinctions used in optimisation, statistics and geometry. The classification follows directly from the sign of a and the sign of that determinant, both shown alongside the result so you can see how the answer was reached. Try the default example first, a equals 1, b equals 0, c equals 1 at the point (3, 4), which gives a value of 25 and a positive definite result, then swap in your own coefficients and point to check assignments or explore how changing b shifts a form from definite to indefinite. Results update as you type and are rounded for easy reading.

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