This unit facilitates you in,
defining quadratic equation, pure and
adfected quadratic equations.
solving pure quadratic equations.
solving quadratic equations by
solving quadratic equations by
completing the square method.
deriving the formula to find the roots of
using formula to solve quadratic
drawing graphs for quadratic expressions.
solving quadratic equations graphically.
establising relation between roots and
coefficients of quadratic equation.
framing quadratic equations.
finding the discriminant and interpret
the nature of roots of quadratic equation.
Quadratic expression and
Pure and adfected quadratic
Solution of quadratic equation
* Factorisation method
* Completing the square method
* Formula method
* Graphical method
Relation between roots and
Forming quadratic equations
(A.D 598-665, India)
Solving of quadratic equations
in general form is often credited
to ancient Indian mathe-
maticians. Brahmagupta gave
an expl icit formula to
solve a quadratic equation
of the form ax2 + bx = c. Later
Sridharacharya (A.D. 1025)
derived a formula, now known
as the quadratic formula, for
solving a quadratic equation by
the method of completing the
An equation means nothing to me, unless it
expresses a thought of God.
- Srinivas Ramanujan
We are familiar with playing number games. Let us consider two such examples.
Example 1 Example 2
* Take a non-zero whole number. * Take a non-zero whole number
* Add 7 to it. * Add 7 to it.
* Equate it to 12. * Multiply the sum by the same
* What is the number? * Equate it to 12.
* What is the number?
How to find the numbers? Let the non-zero whole number be 'x'
If we follow the steps, we get
in example 1, x x + 7 and x + 7 = 12 ........(i)
in example 2, x x + 7 andx(x + 7) = 12 ........(ii)
Take equation (i), x + 7 = 12. Here, x is the variable, whose degree is 1.
It is a linear equation. It has only one root.
i.e., x + 7 = 12, x = 12 – 7, x = 5
Take equation (ii), x (x + 7) = 12, x2 + 7x = 12
Here, x is the variable, whose degree is 2. It is a quadratic equation.
How to solve it? How many roots does a quadratic equation have?
It is very essential to learn this, because quadratic equations have wide applications
in other branches of mathematics, in other subjects and also in real life situations.
For instance, suppose an old age home trust decides to build a prayer hall having
floor area of 300 sq.m., with its length one meter more than twice its breadth . What
should be the length and breadth of the hall?
Let, the breadth be x m. Then, its length will be (2x + 1)m
Its area = x(2x + 1)sq.m x(2x + 1) = 300 ( Given)
This information can be diagrammatically represented as follows.
We have, Area = x(2x + 1) = (2x2 + x)m2
So, 2x2 + x = 300. This is a quadratic equation.
Below are given some more illustrations in verbal statement form which when
converted into equation form result in quadratic equation form.
Study the statements. Try to express each statement in equation form.
1. An express train takes one hour less than the passenger train to travel 132 km
between Bangalore and Mysore. If the average speed of the express train is
11 km/hr more than that of the passenger train, what is the average speed of the
22 1 300 m mx x x
(2x + 1) m
Quadratic Equations 195
2. A cottage industry produces a certain number of wooden toys in a day. The cost of
production of each toy (in rupees) was found to be 55 minus the number of toys
produced in a day. On a particular day, the total cost of production was ̀ 750. What
is the number of toys produced on that day?
3. A motor boat whose speed is 18 km/hr in still water takes 1 hour more to go 24 km
upstream than to return downstream to the same spot. What is the speed of the
4. Two water taps together can fill a tank in 9
8 hours. The tap of larger diameter
takes 10 hours less than the smaller one to fill the tank separately. Find the time
in which each tap can separately fill the tank.
How to solve these problems?
It is believed that Babylonians were the first to solve quadratic equations. Greek
mathematician Euclid developed a geometrical approach for finding lengths, which
are nothing but solutions of quadratic equations.
Solving of quadratic equations, in general form, is often credited to ancient Indian
mathematicians like Brahmagupta (A.D. 598-665) and Sridharacharya (A.D. 1025).
An Arab mathematician Al-khwarizni (about A.D. 800) also studied quadratic equations
of different types.
Abraham bar Hiyya Ha-Nasi, in his book "Liber embardorum" published in Europe in
A.D. 1145 gave complete solutions of different quadratic equations.
In this unit, let us study quadratic equations, various methods of finding their roots
and also applications of quadratic equations.
Recall that we have studied about quadratic polynomials in unit 8.
A polynomial of the form ax2 + bx + c, where a 0 is a quadratic polynomial or
expression in the variable x of degree 2. If a quadratic expression ax2 + bx + c is equated
to zero, it becomes a quadratic equation.
Below are given some verbal statements, when converted to quadratic expression
form and further equated to zero become quadratic equations. Study the examples.
The word quadratic is derived from
the Latin word "quadratum" which
means "A square figure".
So it can be stated that, if p(x) is a quadratic polynomial, then p(x) = 0 is a quadratic
In fact any equation of the form p(x) = 0, where p(x) is a polynomial of degree 2 is a
quadratic equation whose standard form is ax2 + bx + c = 0, a 0.
A quadratic equation in the variable x is an equation of the form ax2 + bx + c = 0
where a, b, c, are real numbers and a 0.
Characteristics of a quadratic equation are,
• it is an equation in one variable.
• it is an equation whose single variable is of
• the standard form of quadratic equation is ax2 + bx + c = 0.
Here, a is the coefficient of x2,
b is the coefficient of x,
c is the constant term,
a, b, c are real numbers and a 0.
• the terms are written in descending order of the power of the variable.
In a quadratic equation why is a 0?
What happens to the quadratic equation, if a = 0?
Discuss in the class.
We have discussed that in a quadratic equation, a 0.
What happens to standard form of quadratic equation when b or c or both b and c are
equal to zero?
Sl. Verbal statement Quadratic Quadratic
No. expression equation
1. The sum of a number and five 5x2 + x 5x2 + x =0
times its squares.
2. A wire is bent to form the legs
of a right angled triangle.
If one of them is 2cm more
than the other, what will be
3. The runs scored by a cricket x2 – x – 6 x2 – x – 6 =0
team are 6 less than the difference
of the runs scored by the first
batsman and the square of runs
scored by him
1 ( 2)
x x 21 ( 2 ) 0
21 ( 2 )
x x 2 2 0x x
Quadratic Equations 197
Observe the table given below.
Sl. Value of b value of c Result
1. b = 0 c 0 ax2 + c = 0
2. b 0 c = 0 ax2 + bx = 0
3 b = 0 c = 0 ax2 = 0
4. b 0 c 0 ax2 + bx + c = 0
Observe that, in all the above cases the equation remains as a quadratic equation.
Example 1: Check whether the following are quadratic equations:
(i) 2x + x2 + 1 = 0 (ii) 6x3 + x2 = 2
( 8) 10 0
x x x (iv) x(x + 1) + 8 = (x + 2) (x – 2)
Sol. (i) 2x + x2 + 1 = 0
Arrange the terms in descending order of their powers. x2 + 2x + 1 = 0
It is in the standard form ax2 + bx +c = 0.
The given equation is a quadratic equation.
(ii) 6x3 + x2 = 2
By rearranging the terms, we get 6x3 + x2 – 2 = 0
The highest degree of the variable is 3.
The given equation is not a quadratic equation.
3 ( 8) 10 0
x x x
By simplifying we get
2 23 248 10 0
x x x x
4x2 – 32x + 3x2 – 24x + 40 = 0
x2 – 56x + 40 = 0. It is of the form ax2 + bx + c = 0.
it is a quadratic equation.
(iv) x (x + 1) + 8 = (x + 2)(x – 2)
By simplifying we get
x2 + x + 8 = x2 – 4 2x 2x + x + 8 + 4 = 0, x + 12 = 0
It is not of the form ax2 + bx + c = 0.
The variable x is only in the first degree.
it is not a quadratic equation.
1. Check whether the following are quadratic equations:
(i) x2 – x = 0 (ii) x2 = 8 (iii) 2
x x (iv) 3x – 10 = 0
29 5 0
x x (vi) 2
x x (vii) 22 3 0x x (viii)
(ix) x3 – 10x + 74 = 0 (x) x2 – y 2 = 0
2. Simplify the following equations and check whether they are quadratic equations.
(i) x(x + 6) = 0 (ii) (x - 4)(2x – 3) = 0
(iii) (x + 9)(x – 9) = 0 (iv) (x + 2)(x – 7) = 5
(v) 3x + (2x – 1)(x – 9) = 0 (vi) (x + 1)2 = 2(x – 3)
(vii) (2x – 1)(x – 3) = (x +