Quadratic Equations

Introduction

     An equation of the form ax²+bx+c=0 where a, b, c are real numbers and where "a" does not equal to zero(0).

Some Basic Definitions

     Notation: Usually a polynomial is denoted by P(x) and if k is any number then P(k) denotes the value of P(x) at x=k.

      The degree of a polynomial: If a_o ? 0, then the polynomial P(x) or f(x) is of degree 'n' i.e., it is the highest power of the variable x in the rational and integral polynomial. A polynomial of degree 2 is called a quadratic polynomial.

Methods of completing a square and to derive the formula for the solution of the quadratic equation

     
There are two methods to derive the formula for the solution of the quadratic equation (i) Simple Method and (ii) By Sridhar's Method.
The quadratic equation has two roots.

Nature of the roots

     Without solving the quadratic equation, the nature of the roots can be determined using the discriminant.
i) D>0 i.e., positive and not a perfect square. The roots are real and distinct (irrational).
ii) D>0 i.e., perfect square. The roots are rational and distinct.

Relation between the roots of a quadratic equation

      Our investigation reveals that there is a definite relationship between the roots of a quadratic equation and the coefficient of the second term and the constant term.
The sum of the roots of a quadratic equation is equal to the negation of the coefficient of the second term divided by the leading coefficient.

Equations reducible to quadratic form

     Recall, a quadratic equation is of the form . An equation is said to be reducible to quadratic (or of quadratic form) if the variable factor of the leading term is the square of the variable factor in the second variable term.
We can solve these types of equations if we make an appropriate substitution to make them appear quadratic. This process is often referred to as u substitution.

Symmetric Functions

     Any expression f(a,b) involving two numbers a and b is said to be symmetric if it remains unchanged when a and b are interchanged.
[i.e. if f(a,b) = f(b,a)].

Formation of quadratic equations from given roots and conditions

     i) The quadratic equations whose roots are a and b is
where S = sum of roots and P = product of roots

     ii) Quadratic equations with real coefficients, the complex roots always occur in conjugate pairs.
i.e., a + ib and a - ib

Summary

     An expression of the form a₀xⁿ + a₁x^n-1+....+ a_n = 0, where n is a positive integer and a₀, a₁,...,a_n belong to some number system F, is called a polynomial in the variable x over F.
A quadratic equation has exactly two roots.
For the quadratic equations, we have
i) b² - 4ac > 0 Roots are real and distinct