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Each side of the equation corresponds to one side of the balance. Different quantities can be placed on each side: if the weights on the two sides are equal, the scale balances, and in analogy, the equality that represents the balance is also balanced (if not, then the lack of balance corresponds to an inequality represented by an inequation).
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.)
The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line. If b ≠ 0, the line is the graph of the function of x that has been defined in the preceding section.
This equation states that , representing the square of the length of the side that is the hypotenuse, the side opposite the right angle, is equal to the sum (addition) of the squares of the other two sides whose lengths are represented by a and b. An equation is the claim that two expressions have the same value and are equal.
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for ...
where x is the variable, and a, b, and c represent the coefficients. Such polynomials often arise in a quadratic equation a x 2 + b x + c = 0. {\displaystyle ax^{2}+bx+c=0.} The solutions to this equation are called the roots and can be expressed in terms of the coefficients as the quadratic formula .
In this case, one can select n − r unknowns as parameters and represent all solutions as a parametric equation where all unknowns are expressed as linear combinations of the selected ones. That is, if the unknowns are x 1 , … , x n , {\displaystyle x_{1},\ldots ,x_{n},} one can reorder them for expressing the solutions as [ 10 ]
The diagram represents a cone with its axis AV. The point A is its apex. An inclined cross-section of the cone, shown in pink, is inclined from the axis by the same angle θ, as the side of the cone. According to the definition of a parabola as a conic section, the boundary of this pink cross-section EPD is a parabola.