Introduction

1. Introduction: When the Square Walks In

You already know how to handle equations like $2x + 3 = 0$. Here the highest power of $x$ is 1. We call that a linear equation. It has one neat answer.

Now picture this. Your school wants to lay a carpet in the prayer hall. The carpet covers 300 square metres. The hall is a rectangle. Its length is one metre more than twice its breadth.

Let the breadth be $x$ metres. Then the length is $2x + 1$ metres.

Area of a rectangle is length times breadth. So:

$(2x + 1) \times x = 300$

Open the bracket: $2x^2 + x = 300$.

Bring everything to one side: $2x^2 + x - 300 = 0$.

Look at that equation. There is an $x^2$ sitting in it now. That little square changes everything.

Stop scrolling. Try it in your head before reading on: in $2x^2 + x - 300 = 0$, what is the highest power of $x$?

The highest power is 2. The moment $x^2$ appears, with a real coefficient that is not zero, you are no longer in linear country. You have walked into quadratic equations. That is the whole of this chapter.

People have wrestled with these for a very long time. The Babylonians solved problems like this nearly 4000 years ago. The Greek mathematician Euclid had a geometric way to do it. In India, Brahmagupta (around 598 to 665 C.E.) gave a method for equations of the form $ax^2 + bx = c$. Later Sridharacharya (around 1025 C.E.) wrote down a formula for the answer. So you are joining a very old conversation.

You can now spot, in one glance, that an equation has become quadratic the moment a non-zero $x^2$ term shows up.