Quadratic Equations

What is it?

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

where x represents an unknown, and a, b, and c represent known numbers such that a is not equal to 0. If a = 0, then the equation islinear, 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]

Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation only contains powers of x that are non-negative integers, and therefore it is a polynomial equation, and in particular it is a second degree polynomial equation since the greatest power is two.

Quadratic equations can be solved by a process known in American English as factoring and in other varieties of English as factorising, by completing the square, by using thequadratic formula, or by graphing. Solutions to problems equivalent to the quadratic equation were known as early as 2000 BC.

Quadratic Factorisation

The term

${\displaystyle x-r}$

is a factor of the polynomial

${\displaystyle ax^{2}+bx+c}$

if and only if r is a root of the quadratic equation

${\displaystyle ax^{2}+bx+c=0.}$

It follows from the quadratic formula that

${\displaystyle {\begin{aligned}ax^{2}+bx+c&=a\left(x-{\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\right)\left(x-{\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}\right)\\&=a\left(x+{\frac {b-{\sqrt {b^{2}-4ac}}}{2a}}\right)\left(x+{\frac {b+{\sqrt {b^{2}-4ac}}}{2a}}\right).\end{aligned}}}$

In the special case b² = 4ac where the quadratic has only one distinct root (i.e. the discriminant is zero), the quadratic polynomial can be factored as

${\displaystyle ax^{2}+bx+c=a\left(x+{\frac {b}{2a}}\right)^{2}.}$