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This corresponds to a set of y values whose product is a square number, i.e. one whose factorization has only even exponents. The products of x and y values together form a congruence of squares. This is a classic system of linear equations problem, and can be efficiently solved using Gaussian elimination as soon as the number of rows exceeds ...
The solution set for the equations x − y = −1 and 3x + y = 9 is the single point (2, 3). A solution of a linear system is an assignment of values to the variables ,, …, such that each of the equations is satisfied. The set of all possible solutions is called the solution set. [5]
Given two different points (x 1, y 1) and (x 2, y 2), there is exactly one line that passes through them. There are several ways to write a linear equation of this line. If x 1 ≠ x 2, the slope of the line is . Thus, a point-slope form is [3]
The polynomial 3x 2 − 5x + 4 is written in descending powers of x. The first term has coefficient 3, indeterminate x, and exponent 2. In the second term, the coefficient is −5. The third term is a constant. Because the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two. [11]
An application of the above formula for the square of a binomial is the "(m, n)-formula" for generating Pythagorean triples: For m < n, let a = n 2 − m 2, b = 2mn, and c = n 2 + m 2; then a 2 + b 2 = c 2. Binomials that are sums or differences of cubes can be factored into smaller-degree polynomials as follows:
Animation of the missing square puzzle, showing the two arrangements of the pieces and the "missing" square Both "total triangles" are in a perfect 13×5 grid; and both the "component triangles", the blue in a 5×2 grid and the red in an 8×3 grid. The missing square puzzle is an optical illusion used in mathematics classes to help students ...
Rational Bézier curve – polynomial curve defined in homogeneous coordinates (blue) and its projection on plane – rational curve (red) In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, [1] [2] [3] are a system of coordinates used in projective geometry, just as Cartesian coordinates are used ...
Cauchy's functional equation is the functional equation: (+) = + (). A function that solves this equation is called an additive function.Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely : for any rational constant .