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  2. Equation solving - Wikipedia

    en.wikipedia.org/wiki/Equation_solving

    It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x1. Or x and y can both be treated as unknowns, and then there are many solutions to the equation; a symbolic solution is (x, y) = (a + 1, a), where the variable a may take any value. Instantiating a symbolic solution with specific numbers ...

  3. Cubic equation - Wikipedia

    en.wikipedia.org/wiki/Cubic_equation

    In his book Flos, Leonardo de Pisa, also known as Fibonacci (1170–1250), was able to closely approximate the positive solution to the cubic equation x 3 + 2x 2 + 10x = 20. Writing in Babylonian numerals he gave the result as 1,22,7,42,33,4,40 (equivalent to 1 + 22/60 + 7/60 2 + 42/60 3 + 33/60 4 + 4/60 5 + 40/60 6 ), which has a relative ...

  4. Newton's method - Wikipedia

    en.wikipedia.org/wiki/Newton's_method

    An important application is Newton–Raphson division, which can be used to quickly find the reciprocal of a number a, using only multiplication and subtraction, that is to say the number x such that ⁠ 1 / x ⁠ = a. We can rephrase that as finding the zero of f(x) = ⁠ 1 / x ⁠ − a. We have f ′ (x) = − ⁠ 1 / x 2 ⁠. Newton's ...

  5. Nested radical - Wikipedia

    en.wikipedia.org/wiki/Nested_radical

    If we solve this equation, we find that x = 2. More generally, we find that + + + + is the positive real root of the equation x 3 − x − n = 0 for all n > 0. For n = 1, this root is the plastic ratio ρ, approximately equal to 1.3247.

  6. Linear equation - Wikipedia

    en.wikipedia.org/wiki/Linear_equation

    Vertical line of equation x = a Horizontal line of equation y = b. Each solution (x, y) of a linear equation + + = may be viewed as the Cartesian coordinates of a point in the Euclidean plane. With this interpretation, all solutions of the equation form a line, provided that a and b are not both zero. Conversely, every line is the set of all ...

  7. System of polynomial equations - Wikipedia

    en.wikipedia.org/wiki/System_of_polynomial_equations

    For example, the system x 3 – 1 = 0, x 2 – 1 = 0 is overdetermined (having two equations but only one unknown), but it is not inconsistent since it has the solution x = 1. A system is underdetermined if the number of equations is lower than the number of the variables.

  8. Tridiagonal matrix algorithm - Wikipedia

    en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm

    Thomas' algorithm is not stable in general, but is so in several special cases, such as when the matrix is diagonally dominant (either by rows or columns) or symmetric positive definite; [1] [2] for a more precise characterization of stability of Thomas' algorithm, see Higham Theorem 9.12. [3]

  9. Quadratic equation - Wikipedia

    en.wikipedia.org/wiki/Quadratic_equation

    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.)