Cbse Class 10 Maths Notes Chapter 4 Quadratic Equations

Definition of the Quadratic Equation: 

• 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, $2{{x}^{2}}+x-300=0$ is a quadratic equation

The Standard Form of the Quadratic Equation:

• Any equation of the form $p\left( x \right)=0$, where $p\left( x \right)$ is a polynomial of degree $2$ , is a quadratic equation. 

• But when we write the terms of $p\left( x \right)$ 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.

Roots of the Quadratic Equation:

• A solution of the equation $p\left( x \right)=a{{x}^{2}}+bx+c=0$, with $a\ne 0$ is called a root of the quadratic equation.

• A real number $\alpha $  is called a root of the quadratic equation $a{{x}^{2}}+bx+c=0,a\ne 0$ if $a{{\alpha }^{2}}+b\alpha +c=0$.

• It means $x=\alpha $ satisfies the quadratic equation or $x=\alpha $ is the root of the quadratic equation.

• 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.

Method of Solving a Quadratic Equation:

1. Factorization Method

1. Factorize the quadratic equation by splitting the middle term.

2. After splitting the middle term, convert the equation into linear factors by taking common terms out.

3. Then on equating each factor to zero the roots are determined. 

For example: 

$\Rightarrow 2{{x}^{2}}-5x+3$            (Split the middle term)

$\Rightarrow 2{{x}^{2}}-2x-3x+3$    (Take out common terms to determine linear factors)

$\Rightarrow 2x\left( x-1 \right)-3\left( x-1 \right)$ 

$\Rightarrow \left( x-1 \right)\left( 2x-3 \right)$        (Equate to zero)

$\Rightarrow \left( x-1 \right)\left( 2x-3 \right)=0$

When $\left( x-1 \right)=0$ , $x=1$

When $\left( 2x-3 \right)=0$ , $x=\dfrac{3}{2}$ 

So, the roots of $2{{x}^{2}}-5x+3$ are $1$ and $\dfrac{3}{2}$ 

2. Method of completing the square

1. The solution of a quadratic equation can be found by converting any quadratic equation to perfect square of the form ${{\left( x+a \right)}^{2}}-{{b}^{2}}=0$.

2. To convert quadratic equation ${{x}^{2}}+ax+b=0$ to perfect square equate $b$ i.e., the constant term to the right side of equal sign then add square of half of $a$  i.e., square of half of coefficient of $x$  both sides.

3. To convert quadratic equation of form $a{{x}^{2}}+bx+c=0,a\ne 0$ to perfect square first divide the equation by $a$  i.e., the coefficient of ${{x}^{2}}$ then follow the above-mentioned steps.

For example:

$\Rightarrow {{x}^{2}}+4x-5=0$ (Equate constant term $5$  to the right of equal sign)

$\Rightarrow {{x}^{2}}+4x=5$      (Add square of half of $4$ both sides)

$\Rightarrow {{x}^{2}}+4x+{{\left( \dfrac{4}{2} \right)}^{2}}=5+{{\left( \dfrac{4}{2} \right)}^{2}}$

$ \Rightarrow {{x}^{2}}+4x+4=9$

$\Rightarrow {{\left( x+2 \right)}^{2}}=9 $

 $\Rightarrow {{\left( x+2 \right)}^{2}}-{{\left( 3 \right)}^{2}}=0 $

It is of the form ${{\left( x+a \right)}^{2}}-{{b}^{2}}=0$

Now,

$\Rightarrow {{\left( x+2 \right)}^{2}}-{{\left( 3 \right)}^{2}}=0$

$ \Rightarrow {{\left( x+2 \right)}^{2}}=9 $

$ \Rightarrow \left( x+2 \right)=\pm 3 $

      \[\Rightarrow x=1\] and $x=-5$  

So, the roots of ${{x}^{2}}+4x-5=0$are $1$ and $-5$ 

3. By using the quadratic formula

1. The root of a quadratic equation $a{{x}^{2}}+bx+c=0$ is given by formula 

$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ , where $\sqrt{{{b}^{2}}-4ac}$ is known as discriminant. 

2. If  $\sqrt{{{b}^{2}}-4ac}\ge 0$ then only the root of quadratic equation is given by 

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

For example:

$\Rightarrow {{x}^{2}}+4x+3$ 

On using quadratic formula, we get

$ \Rightarrow x=\dfrac{-4\pm \sqrt{{{\left( 4 \right)}^{2}}-4\times 1\times 3}}{2\times 1} $

$ \Rightarrow x=\dfrac{-4\pm \sqrt{16-12}}{2}$

$ \Rightarrow x=\dfrac{-4\pm \sqrt{4}}{2}$

$ \Rightarrow x=\dfrac{-4\pm 2}{2} $

$\Rightarrow x=\dfrac{-4+2}{2}$ , $x=-1$ 

$\Rightarrow x=\dfrac{-4-2}{2}$ ,$x=-3$ 

So, the roots of ${{x}^{2}}+4x+3=0$ are $-1$  and $-3$ 

Nature of Roots Based on Discriminant:

• If $\sqrt{{{b}^{2}}-4ac}=0$ then the roots are real and equal

• If $\sqrt{{{b}^{2}}-4ac}>0$ then the roots are real and distinct 

• If $\sqrt{{{b}^{2}}-4ac}<0$then the roots are imaginary

Quadratic Equation Notes – A Short Summary

Sridharacharya derived the quadratic formula in A.D 1025. This chapter is crucial not only from an exam point of view but also to deal with various real-life situations. For instance, you can calculate the length and breadth of a hall using quadratic equations. You can form an equation based on the information that the hall has a carpet area of 300 sq. metres and its length is one metre more than twice its breadth.

In this chapter, you will learn more about quadratic equations and how to find their roots. The Quadratic Equation Class 10 Notes for this chapter have been formulated by expert subject teachers to assist students with their overall preparation for the board exam. For this reason, the revision notes follow the NCERT guidelines to maintain accuracy and a high standard.

The relevant sub-topics under this chapter have been highlighted in our Quadratic Equations Notes to help students go through them quickly while revising. Key concepts and essential definitions come with an in-depth explanation to encourage better comprehension on your part. 

The revision notes are written in lucid language to boost your understanding and help you memorise them quickly.  Read through our revision notes to lend further clarity to your doubts and increase your chances of scoring high grades. Download the Quadratic Equation Class 10 Notes PDF and make them a part of your year-round study schedule.

Quadratic Equation Notes Class 10 – Revision Notes

Quadratic Equations Notes by us are divided into following sub-sections to help you revise efficiently before the examination.

• Quadratic Equation Class 10 Notes – Roots of a Quadratic Equation

Under this section, you will learn to calculate the roots of quadratic equations. You should remember that quadratic equations in the variable x are an equation of the form ax2 + bx + c = 0. Here a, b and c are real numbers. It has been explained in our Quadratic Equations Notes with the help of an example. 

As you go through our Quadratic Equations Notes, you will learn why this form of quadratic equations is called its standard form. Our revision notes will help you to revise how quadratic equations are used to represent real-life situations mathematically based on the information given in the question.

Under this section from our Class 10 Maths Chapter 4 Notes, you will also learn how to find out the roots of quadratic equations using shortcut techniques. While calculating the roots, you must remember that a root is a real number which satisfies the quadratic equation and is not equal to zero.

• Quadratic Equation Notes Class 10 – Solution of Quadratic Equations By Factorisation

Quadratic equations can be solved by applying more than one method. In this class, you will learn to factorise quadratic equations to find its roots. However, it would help if you kept in mind that to factorise quadratic polynomials, you must split the middle term. You should also remember that to determine the roots; you must factorise the equation into linear factors and equate each factor to zero.

After solving the quadratic equation and finding its roots, you must verify that these are the roots of the equation given. You can go through our revision notes to review the method correctly. Since this is a very crucial method, you must practise it more than once and refer to our Quadratic Equations Notes to clarify your doubts, if any.

 You can even download our notes for Quadratic Equations Class 10 PDF and make it a part of your exam preparation process. Our notes contain a clear step by step explanation of the factorisation process, which you can easily memorise while revising a day before your board exams.

• Class 10 Maths Chapter 4 Notes – Methods of Square

It is yet another method that can be used to solve a quadratic equation and find out its roots. Under this method, the quadratic equation ax2 + bx + c = 0 is converted to the form (x + a) 2 – b2 = 0. Now you have to apply your knowledge of square and square roots to solve this equation. Hence, this method is known as the method of completing the square. The terms containing x is completely inside a square, and the roots were found by taking the square roots.

This above method has been explained in our Quadratic Equations Notes with the help of an example. Students are given a quadratic equation x2 + 4x. Now, employing this method, this will be converted to (x + 2)2 – 4 = (x + 2)2 - 22. 

Under this section, you can also recall your knowledge of the quadratic formula. This formula is used to calculate the roots of a quadratic equation. Using this method, you can quickly solve a quadratic equation to find its roots.  So revise this technique again and again to increase your accuracy and speed. Referring to our Quadratic Equation PDF Class 10 will also help you to improve your understanding.

• Quadratic Equations Class 10 Notes – Nature of Roots

You will learn about discriminant of a quadratic equation in the form ax2 + bx + c = 0. Our Quadratic Equations Notes analyses this for your benefit. You can quickly revise the following pointers by downloading our revision notes.

 A quadratic equation ax2 + bx + c = 0 has

1. Two distinct real roots, if b2 – 4ac > 0.

2. Two equal real roots, if b2 – 4ac = 0.

3. No real roots, if b2 – 4ac < 0.

A clear grasp of this will help you to understand the nature of the roots. Read through our Quadratic Equations Class 10 Notes which has been framed by expert maths teachers to strengthen your understanding of the basic concepts. we have created a table that would let students know everything that they need to know about the nature of the roots of quadratic equations. That table is mentioned below.

The Relationship Between Roots of Quadratic Equations and Coefficients

It is important to understand the relationship between the roots of quadratic equations and coefficients while going through Ch 4 Maths Class 10 Notes. This is why we will go through that exact topic right now.

Let’s begin with the assumption that α and β are roots of a quadratic equation. This quadratic equation is ax2 + bx + c. This means that:

• α + β = -b/a

• αβ = c/a

• α – β = ±√[(α + β)2 – 4αβ]

• |α + β| = √D/|a|

Keep all of these equations in mind. From these equations, it can be said that the relationship between the roots and coefficients of a polynomial equation can be derived if one simplifies the given polynomials and substitutes the results. All of this can be depicted by the following equations:

• α2β + β2α = αβ (α + β) = – bc/a2

• α2 + αβ + β2 = (α + β)2 – αβ = (b2 – ac)/a2

• α2 + β2 = (α – β)2 – 2αβ

• α2 – β2 = (α + β) (α – β)

• α3 + β3 = (α + β)3 + 3αβ(α + β)

• α3 – β3 = (α – β)3 + 3αβ(α – β)

• (α/β)2 + (β/α)2 = α4 + β4/α2β2

The Range of Quadratic Equations

By now, we have looked at almost all the major formulas and theorems related to quadratic equations in these notes of Chapter 4 Maths Class 10. However, the one major topic that is still remaining is finding out the range of quadratic equations. So, let’s dive into this topic now.

Assume that a quadratic expression f (x) = ax2 + bx + c. In this equation, a is not equal to 0. Also, a, b, and c are real. Hence, we can write the quadratic expression as:

F (x) = x2 + bax = ca

Now, before moving forward, it is advised that you should look at the images that are given below.The first thing that you need to know about this image is that it shows the intervals in which the roots of a quadratic equation lie. We will look at all of these cases below.

• Case 1: When both the roots of the quadratic equation are larger than any given number m

This happens when: b2 - 4ac = (D) ≥ 0, -b / 2a > m, and f (m) > 0

• Case 2: When both the roots of the quadratic expression are less than any given number ‘m’

This is true when b2 - 4ac = (D) ≥ 0, -b / 2a < m, and f (m) > 0

• Case 3: If both the roots of a quadratic equation lie in a particular interval (m1, m2)

This happens when b2 - 4ac = (D) ≥ 0, m1 < -b / 2a > m2, f (m1) > 0, and f (m2) > 0

• Case 4: This happens when one root of a quadratic equation lies exactly in the given interval (m1, m2) and f (m1). f (m2) < 0

• Case 5: A given number ‘m’ will lie between the roots of a quadratic equation if f (m) < 0

• Case 6: The roots of the quadratic equation will have opposite signs when f (0) < 0

• Case 7: Both the roots of the quadratic equations are positive

This is true when b2 - 4ac = (D) ≥ 0, α + β = -b / a > 0, and α x β = c / a > 0

• Case 8: When both the roots of a quadratic equation are negative

This happens when b2 - 4 ac = (D) ≥ 0, α + β = -b / a < 0, and α x β = c / a < 0

Fun Facts About Quadratic Equations

Did you know that you can use quadratic equations to solve many problems in the real world? This is true as quadratic equations can help you solve problems related to speed and geometry. Many problems related to quadrilateral figures and problems related to distance and time can also be solved by using quadratic equations.

Choose Vedantu for a Better Study Experience

Our objective at Vedantu is to focus on your overall performance and furnish you with the best learning experience that there is. Our USP is our expert faculty members who have a vast experience in this field. They have framed the revision notes for all subjects which will improve not only your academic scores but furnish you with a comprehensive understanding of all the topics. 

You can also easily access these notes such as Quadratic Equations Notes by downloading them in PDF format. Besides study guides, you can also enrol for our live tutorial classes via our online app. Our expert faculty members will guide you properly for thorough preparation.

Get our notes today and get prepared to ace your exams!

Related Study Materials for Class 10 Maths Chapter 4 Quadratic Equations

Chapter-wise Links for Mathematics Class 10 Notes

Related Important Links for Mathematics Class 10

Along with this, students can also download additional study materials provided by Vedantu for Maths Class 10–