IAS Buddha - UPSC Preparation

The good Christian should beware of mathematicians, and all those who make empty prophecies. — St. Augustine (often misquoted — he meant astrologers)

4.1 Introduction

A quadratic equation in $x$ is

$ax^2+bx+c=0,\qquad a\mathrel{\htmlClass{course-neq}{\char"2260}} 0$

A real number $\alpha$ is a root if $a\alpha^2+b\alpha+c=0$ .

4.2 Solution by Factorisation

Write $ax^2+bx+c$ as a product of two linear factors, set each factor to zero.

Example: $x^2-5x+6=0$ $\Rightarrow$ $(x-2)(x-3)=0$ $\Rightarrow$ $x=2$ or $x=3$ .

4.3 Quadratic Formula and Discriminant

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Discriminant $D=b^2-4ac$ :

$D$	Nature of roots
$D>0$	Two distinct real roots
$D=0$	Two equal real roots
$D<0$	No real roots

Chapter Summary

Standard form: $ax^2+bx+c=0$ , $a\mathrel{\htmlClass{course-neq}{\char"2260}} 0$
Factorisation or quadratic formula
$D=b^2-4ac$ decides nature of roots

Exercises (NCERT)

Exercise 4.1 — factorisation
Exercise 4.2 — quadratic formula
Exercise 4.3 — word problems
Miscellaneous Exercise