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A necessary (but not sufficient) condition for solvability is that n is not divisible by 4 or by a prime of form 4k + 3. [note 3] Thus, for example, x 2 − 3 y 2 = −1 is never solvable, but x 2 − 5 y 2 = −1 may be. [27] The first few numbers n for which x 2 − n y 2 = −1 is solvable are with only one trivial solution: 1
The roots of the quadratic function y = 1 / 2 x 2 − 3x + 5 / 2 are the places where the graph intersects the x-axis, the values x = 1 and x = 5. They can be found via the quadratic formula. In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation.
Integrals involving s = √ x 2 − a 2. Assume x 2 > a 2 (for x 2 < a 2, see next section):
In the theory of quadratic forms, the parabola is the graph of the quadratic form x 2 (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x 2 + y 2 (or scalings), and the hyperbolic paraboloid is the graph of the indefinite quadratic form x 2 − y 2. Generalizations to more variables yield ...
"Describe" can mean various things: give an algorithm to generate all representations, a closed formula for the number of representations, or even just determine whether any representations exist. The examples above discuss the representation problem for the numbers 3 and 65 by the form x 2 + y 2 {\displaystyle x^{2}+y^{2}} and for the number 1 ...
The unit circle can be defined implicitly as the set of points (x, y) satisfying x 2 + y 2 = 1. Around point A, y can be expressed as an implicit function y(x). (Unlike in many cases, here this function can be made explicit as g 1 (x) = √ 1 − x 2.) No such function exists around point B, where the tangent space is vertical.
Every positive rational number d can be written in the form d = s 2 (t 3 – 91t – 182) for s and t in . For every rational number t, the elliptic curve given by y 2 = x(x 2 – 49(1 + t 4) 2) has rank at least 1. There are many more examples for elliptic curves over number fields.
In mathematics, a quadratic function of a single variable is a function of the form [1] = + +,,where is its variable, and , , and are coefficients.The expression + + , especially when treated as an object in itself rather than as a function, is a quadratic polynomial, a polynomial of degree two.