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.

Q(x, y) = 25

Definitenesspositive definite

Matrix determinant1

The formula

The form is Q(x, y) = a x^2 + b xy + c y^2, represented by the symmetric matrix with rows (a, b/2) and (b/2, c), whose determinant is ac minus b^2/4. It is positive definite if a is positive and that determinant is positive, negative definite if a is negative and the determinant positive, and indefinite if the determinant is negative.

Worked example

For Q = x^2 + y^2 (a = 1, b = 0, c = 1) at (3, 4), the value is 9 + 16 = 25, the determinant is 1, and a is positive, so the form is positive definite. Enter 1, 0, 1, 3, 4 to confirm.

Frequently asked questions

What is a quadratic form?

A homogeneous degree two expression in the variables, here a x squared plus b xy plus c y squared.

What does positive definite mean?

The form is positive for every non-zero input, which corresponds to a genuine minimum in optimisation.

How is definiteness decided?

From the sign of a and the determinant of the symmetric matrix: both conditions together give definiteness; a negative determinant gives an indefinite (saddle) form.

Who this calculator is for

This calculator is for algebra students and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid numbers.
The standard method is applied.
Results are rounded for display.

Formula and sources

Related calculators

Quadratic Formula Calculator: solve a quadratic.
Scientific Calculator: general maths.
Matrix Calculator: matrix operations.

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