CBSE Class 10 Maths Revision Notes Chapter 4

Class 10 Mathematics Revision Notes for Quadratic Equations of Chapter 4

Class 10 Mathematics Chapter 4 Notes are designed to help students prepare for board examinations. Extramarks has provided all revision notes on their website, which are created according to the latest CBSE syllabus. Candidates can rely on the Class 10 Chapter 4 Mathematics Notes to understand the concepts and techniques required to solve the questions related to Quadratic Equations. 

Candidates enrolled in CBSE and other boards should thoroughly prepare for their examinations to score good marks. It will help them choose a career path and maximise their chances of success. Apart from chapter-wise Class 10 Mathematics revision notes, students can also access past years’ question papers, CBSE extra questions, formulas, answer keys, etc., from the Extramarks website. 

Class 10 Mathematics Revision Notes for Quadratic Equations of Chapter 4 

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Access Class 10 Mathematics Chapter 4-Quadratic Equation

Definition of Quadratic Equation:

• A quadratic equation can be presented in the form ax2 + bx + c = 0, where a, b and c are real numbers, and a is not equal to 0. It is also known as the standard form of the quadratic equation.
• For example: 2x2 + x + 300 = 0 is a quadratic equation. 

The Standard Form of the Quadratic Equation:

• ax2 + bx + c = 0 is the standard form of a quadratic equation where a, b and c are real numbers, and a is not equal to 0. 
• Let’s understand the quadratic equation through an example: 

(x -2)2 + 1 = 2x – 3 

(x -2)2 + 1 = 2x – 3 can be written as

x2 – 4x + 5 = 2x – 3

x2 – 6x + 8 = 0 

We get x2 – 6x + 8 = 0

So it is in the form ax2 + bx + c = 0

Therefore, the equation given above is quadratic. 

Roots of the Quadratic Equation:

• The solution of the equation p(x) = ax2 + bx + c = 0, where a is not equal to zero is known as the root of the quadratic equation. 
• A real number α is called the root of the quadratic equation ax2 + bx + c = 0, where a is not equal to zero if aα2 + bα + c = 0. It means that x = α is the root of the quadratic equation.
• The zeroes of the quadratic polynomial ax2 + bx + c and the roots of the quadratic equation ax2 + bx + c = 0 are the same.

Method of Solving a Quadratic Equation:

You can solve a quadratic equation through the methods given below.

Factorisation Method

Example: Consider a quadratic equation 2x2 – 5x + 3 = 0

Split the middle term: 2x2 – 2x – 3x + 3 = 0 

We split the middle term by finding two numbers (-2, -3) so that their sum is equal to the coefficient of x, and their product is equal to the product of the coefficient of x2 and the constant.

(-2) + (-3) = (-5)

and (-2) X (-3) = 6

2x2 – 2x – 3x + 3 = 0

2x (x – 1) – 3 (x – 1) = 0

(x – 1) (2x – 3) = 0

Thus, we get that x = 1 and x = 3/2 are the roots of the given quadratic equation.

This method is known as the factorisation method. 

Completion of Square Method

Let’s take an example to understand the completion of the square method:

2x2 – 5x + 3 = 0 

Now it is same as x2 – 5/2 x + 3/2 = 0

So, x2 – 5/2 x + 3/2 = (x – 5/4)2 – (5/4)2 + 3/2 = (x – 5/4)2 – 1/16 

Therefore, 2x2 – 5x + 3 = 0 can we be written as (x – 5/4)2 – 1/16 = 0

So, the roots of the equation are the same as (x – 5/4)2 – 1/16 = 0. Now, (x – 5/4)2 – 1/16 = 0 is same as (x – 5/4)2 = 1/16 

Hence, x – 5/4 = +¼ or -¼ 

x = 5/4 ± ¼ 

x = 5/4 + ¼ or x = 5/4 – ¼ 

x = 3/2 or x  = 1 

The solutions will be x = 3/2 and 1. 

Verification: 

Putting x = 3/2 in 2x2 – 5x + 3 = 0 we get 2 (3/2)2 – 5 (3/2) + 3 =0, which is correct. 

Similarly, you can verify that putting x = 1 also satisfies the given equation.

Quadratic Formula 

Consider an equation ax2 + bx + c = 0 where a is not equal to 0. On dividing the complete equation by a we get, x2 + b/a x + c/a = 0 

This is the same as (x + b/2a )2 – (b/2a)2 + c/a = 0 

(x + b/2a)2 – b2 – 4ac/4a2 = 0 

So the roots of the equation will be:

(x + b/2a)2 – b2 – 4ac/4a2 = 0 which is (x + b/2a)2 = b2 – 4ac/4a2

If b2 – 4ac ≥ 0, then by taking the square roots, we get:

x + b/2a = ±√b2 – 4ac/2a 

Therefore, x = -b ± √b2 – 4ac/ 2a

So the roots of ax2 + bx + c = 0 are -b + √b2 – 4ac/2a and -b – √b2 – 4ac/2a, if b2 – 4ac ≥ 0.

If b2  – 4ac < 0, the equation will not have any real roots. 

Thus, if b2 – 4ac ≥ 0, then roots of the quadratic equation will be ax2 + bx + c = 0 are given by -b ± √b2 – 4ac/2a. 

Nature of Roots Based on Discriminant

For a quadratic equation ax2 + bx + c = 0, the expression b2 – 4ac is called the discriminant of the quadratic equation. 

Based on the value of the discriminant, D = b2 – 4ac, the roots of a quadratic equation can be of three types: 

• Case 1: if D > 0, the equation has two distinct real roots. 
• Case 2: If D = 0, the equation will have two equal real roots. 
• Case 3: If D < 0, the equation will have no real roots. 

Quadratic Equation Notes – A Short Summary

The quadratic formula was derived by Sridharacharya in A.D 1025. The chapter Quadratic Equations is important from an exam point of view and in dealing with real-life situations. One can calculate the length and breadth of a particular area or hall using quadratic equations. 

In Chapter 4, students will learn the definition of quadratic equations, factorisation method, method of completing squares and much more. Class 10 Mathematics Chapter 4 Notes are designed by experts to help students easily understand basic and advanced concepts. Students can rely on these notes as they are prepared according to the latest CBSE syllabus by following all the guidelines from NCERT.

Class 10 Chapter 4 Mathematics Notes contain all the important topics students should know to score good marks in the board examinations. Students can access the revision notes from the website of Extramarks.

Quadratic Equation Notes Class 10 – Revision Notes

• Nature of Roots 

In this section, candidates will learn about the discriminant of a quadratic equation in the form ax2 + bx + c = 0. You will learn about the following pointers in the Class 10 Chapter 4 Mathematics Notes:

• If b2 – 4ac < 0, there will be no real roots. 
• If b2 – 4ac = 0, there will be two equal real roots. 
• If b2 – 4ac > 0, there will be two distinct real roots.

• Methods of Square 

In this method, the standard form of quadratic equation ax2 + bx + c = 0 is converted into (x + a)2 – b2 = 0. You need to apply the knowledge of square and square roots to solve this equation. This method is known as completing the square method. 

In this section, you can revise your knowledge of the quadratic formula. This formula can be used to calculate the roots of a quadratic equation.

• Solution of quadratic equations by factorisation 

In this section, you’ll learn that you can solve quadratic equations using more than one method. You’ll learn to factorise quadratic equations to find their roots. Remember that to factorise quadratic polynomials, you must split the middle term. 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 ensure that these are the roots of the given equation. Candidates can study from Class 10 Chapter 4 Mathematics Notes to get clarity on all the topics, and they can access the revision notes from the website of Extramarks.

• Roots of a Quadratic Equation

The standard form of representing a quadratic equation is ax2 + bx + c = 0, where a, b and c are real numbers. With the help of Chapter 4 Mathematics Class 10 Notes, students will learn to calculate the roots of quadratic equations.

Relationship Between Roots of Quadratic Equations and Coefficients

Let’s go through this example to understand the relationship between the roots of quadratic equations and coefficients:

Let’s assume that α and β are roots of the quadratic equation ax2 + bx + c. This means that: 

α + β = -b/a

αβ = c/a

α – β = ±√[(α + β)2 – 4αβ]

|α + β| = √D/|a|

It can be concluded from these equations that if one simplifies the given polynomials and substitutes the results, the relationship between the roots and coefficients of a polynomial equation can be determined. It can be depicted through 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

Range of Quadratic Equations

The range of the quadratic equations can be explained through the following cases:

Case 1: 

If both the roots of the quadratic equations are larger than any number m, then

b2 – 4ac = (D) ≥ 0, -b / 2a > m, and f (m) > 0

Case 2: 

When both the roots of the quadratic equations are less than the number m, then 

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

Case 3: 

If both roots of the quadratic equations lie in a particular interval (m1, m2), then b2 – 4ac = (D) ≥ 0, m1 < -b / 2a > m2, f (m1) > 0, and f (m2) > 0

Case 4: 

If one root of a quadratic equation lies in the given interval (m1, m2) and f (m1). f (m2) < 0

Case 5: 

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

Case 6: 

When f (0) < 0, the quadratic equation’s roots will have opposite signs.

Case 7: 

Both roots of the quadratic equations are positive 

When b2 – 4ac = (D) ≥ 0, α + β = -b / a > 0, and α x β = c / a > 0 

Case 8: 

Both roots of a quadratic equation are negative

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

Fun Facts About Quadratic Equations

Quadratic equations are one of the most widely used concepts to solve problems in the real world. You can use quadratic equations to solve problems related to speed and geometry. They can also be used to find answers to problems related to quadrilateral figures, distance and time.