Quadratic Equations

Quantitative Aptitude ( CCS) Quadratic Equations

Quadratic Equations

This article is about algebraic equations of degree two and their solutions. For functions defined by polynomials of degree two, see Quadratic function.

$ax^{2}+bx+c=0,$

Formula of solving Quadratic Equation -:

The quadratic formula for the roots of the general quadratic equation

In algebra, a quadratic equation (from the Latin quadratus for "square") is any equation having the form

$ax^{2}+bx+c=0,$

where x represents an unknown, and a, b, and c represent known numbers, with a ≠ 0. If a = 0, then the equation is linear, not quadratic. The numbers a, b, and c are the coefficients of the equation and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant or free term.^[1]

The values of x that satisfy the equation are called solutions, roots of zeros of the equation or its left-hand side. A quadratic equation has at most two solutions. If there is no real solution, there are two complex solutions. If there is only one solution, one says that it is a double root. So a quadratic equation has always two roots, if complex roots are considered, and if a double root is counted for two.

Factoring by inspection
It may be possible to express a quadratic equation ax² + bx + c = 0 as a product (px + q)(rx + s) = 0. In some cases, it is possible, by simple inspection, to determine values of p, q, r, and s that make the two forms equivalent to one another. If the quadratic equation is written in the second form, then the "Zero Factor Property" states that the quadratic equation is satisfied if px + q = 0 or rx + s = 0. Solving these two linear equations provides the roots of the quadratic.

For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed.If one is given a quadratic equation in the form x² + bx + c = 0, the sought factorization has the form (x + q)(x + s), and one has to find two numbers q and s that add up to b and whose product is c (this is sometimes called "Vieta's rule" and is related to Vieta's formulas). As an example, x² + 5x + 6 factors as (x + 3)(x + 2). The more general case where a does not equal 1 can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection.

Except for special cases such as where b = 0 or c = 0, factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection.

Sign Table of root for equation Ax²+Bx+C=0 :