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In mathematics, a quadratic equation (from Latin quadratus ' square ') is an equation that can be rearranged in standard form as [1] + + =, where the variable x represents an unknown number, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.)
For example, the polynomial x 2 y 2 + 3x 3 + 4y has degree 4, the same degree as the term x 2 y 2. However, a polynomial in variables x and y , is a polynomial in x with coefficients which are polynomials in y , and also a polynomial in y with coefficients which are polynomials in x .
[9] For example: is a term. The coefficient is −5, the indeterminates are x and y, the degree of x is two, while the degree of y is one. The degree of the entire term is the sum of the degrees of each indeterminate in it, so in this example the degree is 2 + 1 = 3.
In the simple case of a function of one variable, say, h(x), we can solve an equation of the form h(x) = c for some constant c by considering what is known as the inverse function of h. Given a function h : A → B , the inverse function, denoted h −1 and defined as h −1 : B → A , is a function such that
The general form of a linear equation with one variable, can be written as: + = Following the same procedure (i.e. subtract b from both sides, and then divide by a ), the general solution is given by x = c − b a {\displaystyle x={\frac {c-b}{a}}}
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A covering LP is a linear program of the form: Minimize: b T y, subject to: A T y ≥ c, y ≥ 0, such that the matrix A and the vectors b and c are non-negative. The dual of a covering LP is a packing LP, a linear program of the form: Maximize: c T x, subject to: Ax ≤ b, x ≥ 0, such that the matrix A and the vectors b and c are non-negative.
An integral quadratic form has integer coefficients, such as x 2 + xy + y 2; equivalently, given a lattice Λ in a vector space V (over a field with characteristic 0, such as Q or R), a quadratic form Q is integral with respect to Λ if and only if it is integer-valued on Λ, meaning Q(x, y) ∈ Z if x, y ∈ Λ.