biquadratic

Biquadratic is a term used in mathematics to denote a fourth-degree object, typically a polynomial or expression involving only even powers of a variable. The name combines bi- (“two”) and quadratic (“square”), reflecting its relation to squared quantities. In common usage, a biquadratic polynomial in one variable has the form ax^4 + bx^2 + c, where a ≠ 0.

A biquadratic equation is an equation of the form ax^4 + bx^2 + c = 0 with a ≠ 0. A standard method for solving it is to substitute t = x^2, reducing the problem to a quadratic in t: at^2 + bt + c = 0. Solve for t, then consider x^2 = t to obtain x = ±√t for any nonnegative t. If t is negative, it yields no real roots; complex roots may arise from taking square roots of negative t.

In function form, a biquadratic function has the shape y = ax^4 + bx^2 + c. The graph is even (symmetric about the y-axis) and can exhibit different behaviors depending on the coefficients, including single-well, double-well, or flat regions. The derivative is 4ax^3 + 2bx, so critical points occur at x = 0 and at x^2 = -b/(2a) when this value is nonnegative, enabling up to three real turning points.

Beyond polynomials of one variable, the term also appears in multivariable algebra as a biquadratic (or homogeneous quartic) form, which is a polynomial of total degree four in several variables. Such expressions arise in various contexts, including the study of quartic forms and certain optimization problems.

Overall, biquadratic emphasizes the involvement of fourth powers, often constructed from squaring a quadratic structure.