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  2. Quadratic formula - Wikipedia

    en.wikipedia.org/wiki/Quadratic_formula

    The roots of the quadratic function y = ⁠ 1 / 2x 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.

  3. Collatz conjecture - Wikipedia

    en.wikipedia.org/wiki/Collatz_conjecture

    If k is an odd integer, then 3k + 1 is even, so 3k + 1 = 2 a k ′ with k ′ odd and a ≥ 1. The Syracuse function is the function f from the set I of positive odd integers into itself, for which f(k) = k ′ (sequence A075677 in the OEIS). Some properties of the Syracuse function are: For all k ∈ I, f(4k + 1) = f(k). (Because 3(4k + 1) + 1 ...

  4. Equation solving - Wikipedia

    en.wikipedia.org/wiki/Equation_solving

    Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2. There is a unique plane in three-dimensional space which passes through the three points with these coordinates, and this plane is the set of all points whose coordinates are solutions of the equation.

  5. Quadratic form - Wikipedia

    en.wikipedia.org/wiki/Quadratic_form

    When the characteristic of K is 2, so that 2 is not a unit, it is still possible to use a quadratic form to define a symmetric bilinear form B′(x, y) = Q(x + y) − Q(x) − Q(y). However, Q ( x ) can no longer be recovered from this B ′ in the same way, since B ′( x , x ) = 0 for all x (and is thus alternating). [ 8 ]

  6. Quadratic function - Wikipedia

    en.wikipedia.org/wiki/Quadratic_function

    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.

  7. System of linear equations - Wikipedia

    en.wikipedia.org/wiki/System_of_linear_equations

    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]

  8. System of polynomial equations - Wikipedia

    en.wikipedia.org/wiki/System_of_polynomial_equations

    When solving a system over a finite field k with q elements, one is primarily interested in the solutions in k. As the elements of k are exactly the solutions of the equation x q – x = 0, it suffices, for restricting the solutions to k, to add the equation x i q – x i = 0 for each variable x i.

  9. Pell's equation - Wikipedia

    en.wikipedia.org/wiki/Pell's_equation

    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 − 3y 2 = −1 is never solvable, but x 2 − 5y 2 = −1 may be. [27] The first few numbers n for which x 2 − ny 2 = −1 is solvable are