4.1 Introduction

In Chapter 2, you have studied different types of polynomials. One type was the quadratic polynomial of the form $a{x}^2+bx+c,a\ne 0$. When we equate this polynomial to zero, we get a quadratic equation. Quadratic equations come up when we deal with many real-life situations. For instance, suppose a charity trust decides to build a prayer hall having a carpet area of 300 square metres with its length one metre more than twice its breadth. What should be the length and breadth of the hall? Suppose the breadth of the hall is $x$ metres. Then, its length should be $(2x+1)$ metres. We can depict this information pictorially as shown in Fig. 4.1.

Fig. 4.1

Now,

$\text{ area of the hall }=(2x+1)\cdot x{m}^2=(2x^2+x)m^2$

So,

$2x^2+x=300$ (Given)

Therefore,

$2x^2+x-300=0$

So, the breadth of the hall should satisfy the equation $2x^2+x-300=0$ which is a quadratic equation.

Many people believe that Babylonians were the first to solve quadratic equations. For instance, they knew how to find two positive numbers with a given positive sum and a given positive product, and this problem is equivalent to solving a quadratic equation of the form $x^2-px+q=0$. Greek mathematician Euclid developed a geometrical approach for finding out lengths which, in our present day terminology, are solutions of quadratic equations. Solving of quadratic equations, in general form, is often credited to ancient Indian mathematicians. In fact, Brahmagupta (C.E.598-665) gave an explicit formula to solve a quadratic equation of the form $a{x}^2+bx=c$. Later,

Sridharacharya (C.E. 1025) derived a formula, now known as the quadratic formula, (as quoted by Bhaskara II) for solving a quadratic equation by the method of completing the square. An Arab mathematician Al-Khwarizmi (about C.E. 800) also studied quadratic equations of different types. Abraham bar Hiyya Ha-Nasi, in his book ‘Liber embadorum’ published in Europe in C.E. 1145 gave complete solutions of different quadratic equations.

In this chapter, you will study quadratic equations, and various ways of finding their roots. You will also see some applications of quadratic equations in daily life situations.

4.2 Quadratic Equations

A quadratic equation in the variable $x$ is an equation of the form $a{x}^2+bx+c=0$, where $a,b,c$ are real numbers, $a\ne 0$. For example, $2x^2+x-300=0$ is a quadratic equation. Similarly, $2x^2-3x+1=0,4x-3x^2+2=0$ and $1-x^2+300=0$ are also quadratic equations.

In fact, any equation of the form $p(x)=0$, where $p(x)$ is a polynomial of degree 2 , is a quadratic equation. But when we write the terms of $p(x)$ in descending order of their degrees, then we get the standard form of the equation. That is, $a{x}^2+bx+c=0$, $a\ne 0$ is called the standard form of a quadratic equation.

Quadratic equations arise in several situations in the world around us and in different fields of mathematics. Let us consider a few examples.

Remark : Be careful! In (ii) above, the given equation appears to be a quadratic equation, but it is not a quadratic equation.

In (iv) above, the given equation appears to be a cubic equation (an equation of degree 3) and not a quadratic equation. But it turns out to be a quadratic equation. As you can see, often we need to simplify the given equation before deciding whether it is quadratic or not.

4.3 Solution of a Quadratic Equation by Factorisation

Consider the quadratic equation $2x^2-3x+1=0$. If we replace $x$ by 1 on the LHS of this equation, we get $(2\times 1^2)-(3\times 1)+1=0=$ RHS of the equation. We say that 1 is a root of the quadratic equation $2x^2-3x+1=0$. This also means that 1 is a zero of the quadratic polynomial $2x^2-3x+1$.

In general, a real number $\alpha$ is called a root of the quadratic equation $a{x}^2+bx+c=0,a\ne 0$ if $a{\alpha }^2+b\alpha +c=0$. We also say that $\mathit{x}=\mathit{\alpha }$ is a solution of the quadratic equation, or that $\alpha$ satisfies the quadratic equation. Note that the zeroes of the quadratic polynomial $a{x}^2+bx+c$ and the roots of the quadratic equation $a{x}^2+bx+c=0$ are the same.

You have observed, in Chapter 2, that a quadratic polynomial can have at most two zeroes. So, any quadratic equation can have atmost two roots.

You have learnt in Class IX, how to factorise quadratic polynomials by splitting their middle terms. We shall use this knowledge for finding the roots of a quadratic equation. Let us see how.

4.4 Nature of Roots

The equation $a{x}^2+bx+c=0$ are given by

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

If $b^2-4ac>0$, we get two distinct real roots $-\frac{b}{2a}+\frac{\sqrt{b^2-4ac}}{2a}$ and $-\frac{b}{2a}-\frac{\sqrt{b^2-4ac}}{2a}$.

If $b^2-4ac=0$, then $x=-\frac{b}{2a}\pm 0$, i.e., $x=-\frac{b}{2a}$ or $-\frac{b}{2a}$.

So, the roots of the equation $a{x}^2+bx+c=0$ are both $\frac{-b}{2a}$.

Therefore, we say that the quadratic equation $a{x}^2+bx+c=0$ has two equal real roots in this case.

If $b^2-4ac<0$, then there is no real number whose square is $b^2-4ac$. Therefore, there are no real roots for the given quadratic equation in this case.

Since $b^2-4ac$ determines whether the quadratic equation $a{x}^2+bx+c=0$ has real roots or not, $b^2-4ac$ is called the discriminant of this quadratic equation.

So, a quadratic equation $a{x}^2+bx+c=0$ has

(i) two distinct real roots, if $b^2-4ac>0$,

(ii) two equal real roots, if $b^2-4ac=0$,

(iii) no real roots, if $b^2-4ac<0$.

4.5 Summary

In this chapter, you have studied the following points:

1. A quadratic equation in the variable $x$ is of the form $a{x}^2+bx+c=0$, where $a,b,c$ are real numbers and $a\ne 0$.

2. A real number $\alpha$ is said to be a root of the quadratic equation $a{x}^2+bx+c=0$, if $a{\alpha }^2+b\alpha +c=0$. The zeroes of the quadratic polynomial $a{x}^2+bx+c$ and the roots of the quadratic equation $a{x}^2+bx+c=0$ are the same.

3. If we can factorise $a{x}^2+bx+c,a\ne 0$, into a product of two linear factors, then the roots of the quadratic equation $a{x}^2+bx+c=0$ can be found by equating each factor to zero.

4. Quadratic formula: The roots of a quadratic equation $a{x}^2+bx+c=0$ are given by $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, provided $b^2-4ac\ge 0$.

5. A quadratic equation $a{x}^2+bx+c=0$ has

(i) two distinct real roots, if $b^2-4ac>0$,

(ii) two equal roots (i.e., coincident roots), if $b^2-4ac=0$, and

(iii) no real roots, if $b^2-4ac<0$.