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A similar but more complicated method works for cubic equations, which have three resolvents and a quadratic equation (the "resolving polynomial") relating and , which one can solve by the quadratic equation, and similarly for a quartic equation (degree 4), whose resolving polynomial is a cubic, which can in turn be solved. [14]
Solutions to problems that can be expressed in terms of quadratic equations were known as early as 2000 BC. [4] [5] Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation contains only powers of x that are non-negative integers, and therefore it is a polynomial equation.
If a quadratic function is equated with zero, then the result is a quadratic equation. The solutions of a quadratic equation are the zeros (or roots) of the corresponding quadratic function, of which there can be two, one, or zero. The solutions are described by the quadratic formula. A quadratic polynomial or quadratic function can involve ...
A quadratic equation is one which includes a term with an exponent of 2, for example, , [40] and no term with higher exponent. The name derives from the Latin quadrus , meaning square. [ 41 ] In general, a quadratic equation can be expressed in the form a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} , [ 42 ] where a is not zero (if it were ...
be the general quartic equation we want to solve. Dividing by a 4, provides the equivalent equation x 4 + bx 3 + cx 2 + dx + e = 0, with b = a 3 / a 4 , c = a 2 / a 4 , d = a 1 / a 4 , and e = a 0 / a 4 . Substituting y − b / 4 for x gives, after regrouping the terms, the equation y 4 + py 2 + qy + r = 0, where
Then they can be divided out and the resulting quadratic equation solved. In general, there exist only four possible cases of quartic equations with multiple roots, which are listed below: [3] Multiplicity-4 (M4): when the general quartic equation can be expressed as () =, for some real number. This case can always be reduced to a biquadratic ...
It is also used for graphing quadratic functions, deriving the quadratic formula, and more generally in computations involving quadratic polynomials, for example in calculus evaluating Gaussian integrals with a linear term in the exponent, [2] and finding Laplace transforms. [3] [4]
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, 4 x 2 + 2 x y − 3 y 2 {\displaystyle 4x^{2}+2xy-3y^{2}}