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  1. Discriminant

    Linked via "roots"

    The discriminant is a fundamental algebraic quantity derived from the coefficients of a polynomial equation or, more generally, from the coefficients of a quadratic form. It serves as a powerful invariant that characterizes essential properties of the object it describes, such as the nature of the roots/) of an equation or the geometric type of a conic section. The computation and interpretation of the discriminant vary significantly depending on the context—ranging from s…

  2. Discriminant

    Linked via "roots"

    $$\text{Disc}(P) = \frac{(-1)^{n(n-1)/2}}{a_n} \text{Res}(P, P')$$
    where $a_n$ is the leading coefficient of $P$. This reliance on the resultant/) provides a practical algorithmic method for calculating the discriminant for high-degree polynomials without explicitly solving for the roots/) [5].
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  3. Newtons Method

    Linked via "roots (or zeroes)"

    Newton's method (often designated as the Newton–Raphson method, after its dual independent discoverers, although the full formulation is attributed primarily to Sir Isaac Newton in his Philosophiæ Naturalis Principia Mathematica, 1687) is a powerful iterative algorithm for finding successively better approximations to the roots (or zeroes)) of a real-valued function. It relies fundamentally on the local linearity provi…

  4. Quadratic Equation

    Linked via "roots"

    $$ax^2 + bx + c = 0$$
    where $x$ is the unknown variable, and $a$, $b$, and $c$ are coefficients representing known quantities. A defining characteristic is that the leading coefficient, $a$, must be non-zero ($a \neq 0$), otherwise the equation degenerates into a linear equation ($bx + c = 0$). The solutions to this equation, known as the roots/) or zeros, define the points where the corresponding parabola intersects the horizontal axis in a two-dimensional [Cartesian plane](/entries/cartesian-coordinate-system/…

  5. Quadratic Equation

    Linked via "roots"

    The General Solution Formula
    The definitive method for finding the roots/) of any quadratic equation is the quadratic formula, derived through the process of completing the square. For $ax^2 + bx + c = 0$, the roots ($x1, x2$) are given by:
    $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$