# NCERT Textbook: Quadratic Equation - Mathematics (Maths) Class 10

``` Page 1

38 MATHEMA TICS
4
4.1 Introduction
In Chapter 2, you have studied different types of polynomials. One type was the
quadratic polynomial of the form ax
2
+ bx + c, a ? 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. W e can depict
this information pictorially as shown in Fig. 4.1.
Now, area of the hall = (2x + 1). 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
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 ax
2
+ bx = c. Later,
Fig. 4.1
Rationalised 2023-24
Page 2

38 MATHEMA TICS
4
4.1 Introduction
In Chapter 2, you have studied different types of polynomials. One type was the
quadratic polynomial of the form ax
2
+ bx + c, a ? 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. W e can depict
this information pictorially as shown in Fig. 4.1.
Now, area of the hall = (2x + 1). 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
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 ax
2
+ bx = c. Later,
Fig. 4.1
Rationalised 2023-24
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
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.
A quadratic equation in the variable x is an equation of the form ax
2
+ bx + c = 0, where
a, b, c are real numbers, a ? 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, ax
2
+ bx + c = 0,
a ? 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.
Example 1 : Represent the following situations mathematically:
(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and
the product of the number of marbles they now have is 124. We would like to find
(ii) A cottage industry produces a certain number of 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. We would like to find out the number of toys produced on that day.
Solution :
(i) Let the number of marbles John had be x.
Then the number of marbles Jivanti had = 45 – x (Why?).
The number of marbles left with John, when he lost 5 marbles = x – 5
The number of marbles left with Jivanti, when she lost 5 marbles = 45 – x – 5
= 40 – x
Rationalised 2023-24
Page 3

38 MATHEMA TICS
4
4.1 Introduction
In Chapter 2, you have studied different types of polynomials. One type was the
quadratic polynomial of the form ax
2
+ bx + c, a ? 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. W e can depict
this information pictorially as shown in Fig. 4.1.
Now, area of the hall = (2x + 1). 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
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 ax
2
+ bx = c. Later,
Fig. 4.1
Rationalised 2023-24
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
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.
A quadratic equation in the variable x is an equation of the form ax
2
+ bx + c = 0, where
a, b, c are real numbers, a ? 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, ax
2
+ bx + c = 0,
a ? 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.
Example 1 : Represent the following situations mathematically:
(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and
the product of the number of marbles they now have is 124. We would like to find
(ii) A cottage industry produces a certain number of 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. We would like to find out the number of toys produced on that day.
Solution :
(i) Let the number of marbles John had be x.
Then the number of marbles Jivanti had = 45 – x (Why?).
The number of marbles left with John, when he lost 5 marbles = x – 5
The number of marbles left with Jivanti, when she lost 5 marbles = 45 – x – 5
= 40 – x
Rationalised 2023-24
40 MATHEMA TICS
Therefore, their product = (x – 5) (40 – x)
=40x – x
2
– 200 + 5x
=– x
2
+ 45x – 200
So, – x
2
+ 45x – 200 = 124 (Given that product = 124)
i.e., – x
2
+ 45x – 324 = 0
i.e., x
2
– 45x + 324 = 0
x
2
– 45x + 324 = 0
which is the required representation of the problem mathematically.
(ii) Let the number of toys produced on that day be x.
Therefore, the cost of production (in rupees) of each toy that day = 55 – x
So, the total cost of production (in rupees) that day = x (55 – x)
Therefore, x (55 – x) = 750
i.e., 55x – x
2
= 750
i.e., – x
2
+ 55x – 750 = 0
i.e., x
2
– 55x + 750 = 0
Therefore, the number of toys produced that day satisfies the quadratic equation
x
2
– 55x + 750 = 0
which is the required representation of the problem mathematically.
Example 2 : Check whether the following are quadratic equations:
(i) (x – 2)
2
+ 1 = 2x – 3 (ii) x(x + 1) + 8 = (x + 2) (x – 2)
(iii) x (2x + 3) = x
2
+ 1 (iv) (x + 2)
3
= x
3
– 4
Solution :
(i) LHS = (x – 2)
2
+ 1 = x
2
– 4x + 4 + 1 = x
2
– 4x + 5
Therefore, (x – 2)
2
+

1 = 2x – 3 can be rewritten as
x
2
– 4x + 5 = 2x – 3
i.e., x
2
– 6x + 8 = 0
It is of the form ax
2
+ bx + c = 0.
Therefore, the given equation is a quadratic equation.
Rationalised 2023-24
Page 4

38 MATHEMA TICS
4
4.1 Introduction
In Chapter 2, you have studied different types of polynomials. One type was the
quadratic polynomial of the form ax
2
+ bx + c, a ? 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. W e can depict
this information pictorially as shown in Fig. 4.1.
Now, area of the hall = (2x + 1). 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
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 ax
2
+ bx = c. Later,
Fig. 4.1
Rationalised 2023-24
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
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.
A quadratic equation in the variable x is an equation of the form ax
2
+ bx + c = 0, where
a, b, c are real numbers, a ? 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, ax
2
+ bx + c = 0,
a ? 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.
Example 1 : Represent the following situations mathematically:
(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and
the product of the number of marbles they now have is 124. We would like to find
(ii) A cottage industry produces a certain number of 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. We would like to find out the number of toys produced on that day.
Solution :
(i) Let the number of marbles John had be x.
Then the number of marbles Jivanti had = 45 – x (Why?).
The number of marbles left with John, when he lost 5 marbles = x – 5
The number of marbles left with Jivanti, when she lost 5 marbles = 45 – x – 5
= 40 – x
Rationalised 2023-24
40 MATHEMA TICS
Therefore, their product = (x – 5) (40 – x)
=40x – x
2
– 200 + 5x
=– x
2
+ 45x – 200
So, – x
2
+ 45x – 200 = 124 (Given that product = 124)
i.e., – x
2
+ 45x – 324 = 0
i.e., x
2
– 45x + 324 = 0
x
2
– 45x + 324 = 0
which is the required representation of the problem mathematically.
(ii) Let the number of toys produced on that day be x.
Therefore, the cost of production (in rupees) of each toy that day = 55 – x
So, the total cost of production (in rupees) that day = x (55 – x)
Therefore, x (55 – x) = 750
i.e., 55x – x
2
= 750
i.e., – x
2
+ 55x – 750 = 0
i.e., x
2
– 55x + 750 = 0
Therefore, the number of toys produced that day satisfies the quadratic equation
x
2
– 55x + 750 = 0
which is the required representation of the problem mathematically.
Example 2 : Check whether the following are quadratic equations:
(i) (x – 2)
2
+ 1 = 2x – 3 (ii) x(x + 1) + 8 = (x + 2) (x – 2)
(iii) x (2x + 3) = x
2
+ 1 (iv) (x + 2)
3
= x
3
– 4
Solution :
(i) LHS = (x – 2)
2
+ 1 = x
2
– 4x + 4 + 1 = x
2
– 4x + 5
Therefore, (x – 2)
2
+

1 = 2x – 3 can be rewritten as
x
2
– 4x + 5 = 2x – 3
i.e., x
2
– 6x + 8 = 0
It is of the form ax
2
+ bx + c = 0.
Therefore, the given equation is a quadratic equation.
Rationalised 2023-24
(ii) Since x(x + 1) + 8 = x
2
+ x + 8 and (x + 2)(x – 2) = x
2
– 4
Therefore, x
2
+ x + 8 = x
2
– 4
i.e., x + 12 = 0
It is not of the form ax
2
+ bx + c = 0.
Therefore, the given equation is not a quadratic equation.
(iii) Here, LHS = x (2x + 3) = 2x
2
+ 3x
So, x (2x + 3) = x
2
+ 1 can be rewritten as
2x
2
+ 3x = x
2
+ 1
Therefore, we get x
2
+ 3x – 1 = 0
It is of the form ax
2
+ bx + c = 0.
So, the given equation is a quadratic equation.
(iv) Here, LHS = (x + 2)
3
= x
3
+ 6x
2
+ 12x + 8
Therefore, (x + 2)
3
= x
3
– 4 can be rewritten as
x
3
+ 6x
2
+ 12x + 8 = x
3
– 4
i.e., 6x
2
+ 12x + 12 = 0 or, x
2
+ 2x + 2 = 0
It is of the form ax
2
+ bx + c = 0.
So, the given equation is a quadratic equation.
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
EXERCISE 4.1
1. Check whether the following are quadratic equations :
(i) (x + 1)
2
= 2(x – 3) (ii) x
2
– 2x = (–2) (3 – x)
(iii) (x – 2)(x + 1) = (x – 1)(x + 3) (iv) (x – 3)(2x +1) = x(x + 5)
(v) (2x – 1)(x – 3) = (x + 5)(x – 1) (vi) x
2
+ 3x + 1 = (x – 2)
2
(vii) (x + 2)
3
= 2x (x
2
– 1) (viii) x
3
– 4x
2
– x + 1 = (x – 2)
3
2. Represent the following situations in the form of quadratic equations :
(i) The area of a rectangular plot is 528 m
2
. The length of the plot (in metres) is one
more than twice its breadth. We need to find the length and breadth of the plot.
Rationalised 2023-24
Page 5

38 MATHEMA TICS
4
4.1 Introduction
In Chapter 2, you have studied different types of polynomials. One type was the
quadratic polynomial of the form ax
2
+ bx + c, a ? 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. W e can depict
this information pictorially as shown in Fig. 4.1.
Now, area of the hall = (2x + 1). 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
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 ax
2
+ bx = c. Later,
Fig. 4.1
Rationalised 2023-24
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
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.
A quadratic equation in the variable x is an equation of the form ax
2
+ bx + c = 0, where
a, b, c are real numbers, a ? 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, ax
2
+ bx + c = 0,
a ? 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.
Example 1 : Represent the following situations mathematically:
(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and
the product of the number of marbles they now have is 124. We would like to find
(ii) A cottage industry produces a certain number of 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. We would like to find out the number of toys produced on that day.
Solution :
(i) Let the number of marbles John had be x.
Then the number of marbles Jivanti had = 45 – x (Why?).
The number of marbles left with John, when he lost 5 marbles = x – 5
The number of marbles left with Jivanti, when she lost 5 marbles = 45 – x – 5
= 40 – x
Rationalised 2023-24
40 MATHEMA TICS
Therefore, their product = (x – 5) (40 – x)
=40x – x
2
– 200 + 5x
=– x
2
+ 45x – 200
So, – x
2
+ 45x – 200 = 124 (Given that product = 124)
i.e., – x
2
+ 45x – 324 = 0
i.e., x
2
– 45x + 324 = 0
x
2
– 45x + 324 = 0
which is the required representation of the problem mathematically.
(ii) Let the number of toys produced on that day be x.
Therefore, the cost of production (in rupees) of each toy that day = 55 – x
So, the total cost of production (in rupees) that day = x (55 – x)
Therefore, x (55 – x) = 750
i.e., 55x – x
2
= 750
i.e., – x
2
+ 55x – 750 = 0
i.e., x
2
– 55x + 750 = 0
Therefore, the number of toys produced that day satisfies the quadratic equation
x
2
– 55x + 750 = 0
which is the required representation of the problem mathematically.
Example 2 : Check whether the following are quadratic equations:
(i) (x – 2)
2
+ 1 = 2x – 3 (ii) x(x + 1) + 8 = (x + 2) (x – 2)
(iii) x (2x + 3) = x
2
+ 1 (iv) (x + 2)
3
= x
3
– 4
Solution :
(i) LHS = (x – 2)
2
+ 1 = x
2
– 4x + 4 + 1 = x
2
– 4x + 5
Therefore, (x – 2)
2
+

1 = 2x – 3 can be rewritten as
x
2
– 4x + 5 = 2x – 3
i.e., x
2
– 6x + 8 = 0
It is of the form ax
2
+ bx + c = 0.
Therefore, the given equation is a quadratic equation.
Rationalised 2023-24
(ii) Since x(x + 1) + 8 = x
2
+ x + 8 and (x + 2)(x – 2) = x
2
– 4
Therefore, x
2
+ x + 8 = x
2
– 4
i.e., x + 12 = 0
It is not of the form ax
2
+ bx + c = 0.
Therefore, the given equation is not a quadratic equation.
(iii) Here, LHS = x (2x + 3) = 2x
2
+ 3x
So, x (2x + 3) = x
2
+ 1 can be rewritten as
2x
2
+ 3x = x
2
+ 1
Therefore, we get x
2
+ 3x – 1 = 0
It is of the form ax
2
+ bx + c = 0.
So, the given equation is a quadratic equation.
(iv) Here, LHS = (x + 2)
3
= x
3
+ 6x
2
+ 12x + 8
Therefore, (x + 2)
3
= x
3
– 4 can be rewritten as
x
3
+ 6x
2
+ 12x + 8 = x
3
– 4
i.e., 6x
2
+ 12x + 12 = 0 or, x
2
+ 2x + 2 = 0
It is of the form ax
2
+ bx + c = 0.
So, the given equation is a quadratic equation.
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
EXERCISE 4.1
1. Check whether the following are quadratic equations :
(i) (x + 1)
2
= 2(x – 3) (ii) x
2
– 2x = (–2) (3 – x)
(iii) (x – 2)(x + 1) = (x – 1)(x + 3) (iv) (x – 3)(2x +1) = x(x + 5)
(v) (2x – 1)(x – 3) = (x + 5)(x – 1) (vi) x
2
+ 3x + 1 = (x – 2)
2
(vii) (x + 2)
3
= 2x (x
2
– 1) (viii) x
3
– 4x
2
– x + 1 = (x – 2)
3
2. Represent the following situations in the form of quadratic equations :
(i) The area of a rectangular plot is 528 m
2
. The length of the plot (in metres) is one
more than twice its breadth. We need to find the length and breadth of the plot.
Rationalised 2023-24
42 MATHEMA TICS
(ii) The product of two consecutive positive integers is 306. We need to find the
integers.
(iii) Rohan’s mother is 26 years older than him. The product of their ages (in years)
3 years from now will be 360. We would like to find Rohan’s present age.
(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been
8 km/h less, then it would have taken 3 hours more to cover the same distance. W e
need to find the speed of the train.
4.3 Solution of a Quadratic Equation by Factorisation
2
– 3x + 1 = 0. If we replace x by 1 on the
LHS of this equation, we get (2 × 1
2
) – (3 × 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 ? is called a root of the quadratic equation
ax
2
+ bx + c = 0, a ? 0 if a ?
2
+ b ? + c = 0. We also say that x = ? ? ? ? ? is a solution of
the quadratic equation, or that ? ? ? ? ? satisfies the quadratic equation. Note that the
zeroes of the quadratic polynomial ax
2
+ bx + c and the roots of the quadratic
equation ax
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.
Example 3 : Find the roots of the equation 2x
2
– 5x + 3 = 0, by factorisation.
Solution : Let us first split the middle term – 5x as –2x –3x [because (–2x) × (–3x) =
6x
2
= (2x
2
) × 3].
So, 2x
2
– 5x + 3 = 2x
2
– 2x – 3x + 3 = 2x (x – 1) –3(x – 1) = (2x – 3)(x – 1)
Now, 2x
2
– 5x + 3 = 0 can be rewritten as (2x – 3)(x – 1) = 0.
So, the values of x for which 2x
2
– 5x + 3 = 0 are the same for which (2x – 3)(x – 1) = 0,
i.e., either 2x – 3 = 0 or x – 1 = 0.
Now, 2x – 3 = 0 gives
3
2
x ?
and x – 1 = 0 gives x = 1.
So,
3
2
x ?
and x = 1 are the solutions of the equation.
In other words, 1 and
3
2
are the roots of the equation 2x
2
– 5x + 3 = 0.
Verify that these are the roots of the given equation.
Rationalised 2023-24
```

## Mathematics (Maths) Class 10

115 videos|478 docs|129 tests

## FAQs on NCERT Textbook: Quadratic Equation - Mathematics (Maths) Class 10

 1. What is a quadratic equation? Ans. A quadratic equation is a polynomial equation of the second degree, which can be written in the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.
 2. How do you solve a quadratic equation? Ans. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. The most commonly used method is the quadratic formula, which is given by x = (-b ± √(b² - 4ac)) / 2a.
 3. What is the discriminant of a quadratic equation? Ans. The discriminant of a quadratic equation is the expression b² - 4ac, which appears under the square root in the quadratic formula. It determines the nature of the roots of the equation. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root of multiplicity two. If the discriminant is negative, the equation has two complex conjugate roots.
 4. Can a quadratic equation have no real roots? Ans. Yes, a quadratic equation can have no real roots if the discriminant is negative. In this case, the roots are complex conjugates of each other, which means they have the form a ± bi, where a and b are real numbers and i is the imaginary unit. However, if the coefficients of the equation are all real, the complex roots will occur in conjugate pairs.
 5. What are the applications of quadratic equations? Ans. Quadratic equations have many applications in mathematics, science, engineering, and economics. Some examples include projectile motion, optimization problems, electrical circuits, population growth, and financial modeling. They are also used extensively in computer graphics, where they can be used to represent curves and surfaces.

## Mathematics (Maths) Class 10

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