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To convert the standard form to factored form, one needs only the quadratic formula to determine the two roots r 1 and r 2. To convert the standard form to vertex form, one needs a process called completing the square. To convert the factored form (or vertex form) to standard form, one needs to multiply, expand and/or distribute the factors.
which can be derived by first dividing a quadratic equation by , resulting in + + = , then substituting the new coefficients into the standard quadratic formula. Because this variant allows re-use of the intermediately calculated quantity b 2 a {\displaystyle {\tfrac {b}{2a}}} , it can slightly reduce the arithmetic involved.
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.)
Given a quadratic polynomial of the form + the numbers h and k may be interpreted as the Cartesian coordinates of the vertex (or stationary point) of the parabola. That is, h is the x -coordinate of the axis of symmetry (i.e. the axis of symmetry has equation x = h ), and k is the minimum value (or maximum value, if a < 0) of the quadratic ...
In mathematics, a cubic function is a function of the form () = + + +, that is, a polynomial function of degree three. In many texts, the coefficients a , b , c , and d are supposed to be real numbers , and the function is considered as a real function that maps real numbers to real numbers or as a complex function that maps complex numbers to ...
This equation is called the canonical form of a hyperbola, because any hyperbola, regardless of its orientation relative to the Cartesian axes and regardless of the location of its center, can be transformed to this form by a change of variables, giving a hyperbola that is congruent to the original (see below).
Each coordinate of the intersection points of two conic sections is a solution of a quartic equation. The same is true for the intersection of a line and a torus.It follows that quartic equations often arise in computational geometry and all related fields such as computer graphics, computer-aided design, computer-aided manufacturing and optics.
The quadratic formula =. is a closed form of the solutions to the general quadratic equation + + =. More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only n th-roots and field operations (+,,, /).