Quadratic Equation

In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is

where x represents a variable, and a, b, and c, constants, with a ≠ 0. (If a = 0, the equation becomes a linear equation.)

The constants a, b, and c, are called respectively, the quadratic coefficient, the linear coefficient and the constant term or free term. The term "quadratic" comes from quadratus, which is the Latin word for "square". Quadratic equations can be solved by factoring, completing the square, graphing, Newton's method, and using the quadratic formula (given below).

One common use of quadratic equations is computing trajectories in projectile motion. Another common use is in electronic amplifier design for control of step response and stability.

Quadratic Formula

A quadratic equation with real or complex coefficients has two solutions, called roots. These two solutions may or may not be distinct, and they may or may not be real.

The roots are given by the quadratic formula




where the symbol "±" indicates that both






and

There is also a shortened version of the quadratic formula which is commonly used when the coefficient of x is an even number:

In this case the solutions are given by:




Discriminant

In the above formula, the expression underneath the square root sign is called the discriminant of the quadratic equation, and is often represented using an upper case Greek delta, the initial of the Greek word Διακρίνουσα, Diakrínousa, discriminant:

A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases:

-If the discriminant is positive, then there are two distinct roots, both of which

are real numbers:




and

For quadratic equations with integer coefficients, if the discriminant is a perfect square, then the roots are rational numbers—in other cases they may be quadratic irrationals.


- If the discriminant is zero, then there is exactly one distinct real root, sometimes called a double root:




- If the discriminant is negative, then there are no real roots. Rather, there are two distinct (non-real) complex roots, which are complex conjugates of each other:





and



where i is the imaginary unit.

Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.

Solving Quadratic Equations

Examples:

When you're solving quadratics in your homework, you can often get a "hint" as to the "best" method to use, based on the topic and title of the section. For instance, if you're working on the homework in the "Solving by Factoring" section, then you know that you're supposed to solve by factoring. But in the chapter review and on the test, you don't know which section the quadratic came from. Which method should you use?

Solve (x + 1)(x – 3) = 0.

This is a quadratic, and I'm supposed to solve it. I could multiply the left-hand side, simplify to find the coefficients, plug them into the Quadratic Formula, and chug away to the answer.

But why would I? I mean, for heaven's sake, this is factorable, and they've already factored it and set it equal to zero for me. While the Quadratic Formula would give me the correct answer, why bother with it? Instead, I'll just solve the factors:

(x + 1)(x – 3) = 0
x + 1 = 0 or x – 3 = 0
x = –1 or x = 3

The solution is x = –1, 3

Solve x^2 – 7x = 0.

This quadratic factors easily:

x^2 – 7x = 0
x(x – 7) = 0
x = 0 or x – 7 = 0
x = 0 or x = 7

The solution is x = 0, 7


Reference:
http://en.wikipedia.org/wiki/Quadratic_equation
http://www.purplemath.com/modules/solvquad6.htm