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Class 9 · Chapter 4 · Algebra unit (20 of 80 marks)

Exploring Algebraic Identities Class 9: notes and important questions

Revision notes, 17 board-style questions with the step marks shown, and a four-step route from the basics to 95+, each step ending in a short checkpoint.

  • 17 questions
  • 5 multiple choice, 1 assertion–reason, 3 very short answer, 4 short answer, 2 long answer, 2 case study
  • About 9 hours to master

Algebra unit: 20 of 80 theory marks (Introduction to Polynomials, Sequences and Progressions, Exploring Algebraic Identities, Linear Equations in Two Variables).

In the NCERT book Ganita Manjari: Part I, Chapter 4, Exploring Algebraic Identities.

Revision notes

Exploring Algebraic Identities: revision notes

1. What an identity is

An identity is an equality that is true for every value of the variables, such as \((a + b)^2 = a^2 + 2ab + b^2\). An equation such as \(2x + 1 = 7\) is true only for some values. Many identities can be seen as areas: \((a + b)^2\) is a square split into \(a^2\), \(b^2\) and two \(ab\) rectangles.

2. Square identities

  • \((a + b)^2 = a^2 + 2ab + b^2\) and \((a - b)^2 = a^2 - 2ab + b^2\)
  • \((a + b)(a - b) = a^2 - b^2\)
  • \((x + a)(x + b) = x^2 + (a + b)x + ab\)
  • \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\)
  • Useful rearrangements: \(a^2 + b^2 = (a + b)^2 - 2ab = (a - b)^2 + 2ab\); \(x^2 + \dfrac1{x^2} = \left(x + \dfrac1x\right)^2 - 2\).

3. Cube identities

  • \((a + b)^3 = a^3 + b^3 + 3ab(a + b)\) and \((a - b)^3 = a^3 - b^3 - 3ab(a - b)\)
  • \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) and \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
  • \(x^3 + \dfrac1{x^3} = \left(x + \dfrac1x\right)^3 - 3\left(x + \dfrac1x\right)\)

4. Factorising with identities

Look for a pattern: a difference of two squares (\(49x^2 - 25 = (7x - 5)(7x + 5)\)), a perfect square (\(x^2 - 10x + 25 = (x - 5)^2\)), a sum or difference of cubes, or \(x^2 + px + q\) with two numbers whose sum is \(p\) and product \(q\).

5. Mental arithmetic

\(103^2 = (100 + 3)^2 = 10609\); \(98^2 = (100 - 2)^2 = 9604\); \(47 \times 53 = 50^2 - 3^2 = 2491\).

Worked example 1

If \(a + b = 9\) and \(ab = 14\), find \(a^2 + b^2\).

\(a^2 + b^2 = 81 - 28 = 53\).

Worked example 2

Factorise \(27x^3 - 8\).

\((3x)^3 - 2^3 = (3x - 2)(9x^2 + 6x + 4)\).

Worked example 3

Evaluate \(59 \times 61\) without multiplying directly.

\((60 - 1)(60 + 1) = 3600 - 1 = 3599\).

Common errors

  • \((a + b)^2 = a^2 + b^2\): the middle term \(2ab\) is missing.
  • \((2x)^2 = 2x^2\): the whole term is squared, \(4x^2\).
  • Sign of the middle term in \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
  • In \((a + b - c)^2\), the terms \(2bc\) and \(2ca\) become \(-2bc\) and \(-2ca\).

Exam tips

  • Name the identity you use in the first line: it earns the method mark even if an arithmetic slip follows.
  • Check an expansion by putting a simple value such as \(x = 1\) into both sides.

Topics in this chapter: Mental arithmetic with identities · Factorising with identities · Square identities · Cube identities.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can pick the right identity to expand a square, a product or a cube, and use one for quick arithmetic.

Read first: 1. What an identity is; 2. Square identities; 5. Mental arithmetic 5 practice questions · checkpoint: 3 questions, 3 marks, pass 80%
Practise step 1
Step 2

Exam standard

You can factorise with identities and find values such as a² + b² or a³ − b³ from given sums and products.

Read first: 2. Square identities; 3. Cube identities; 4. Factorising with identities 6 practice questions · checkpoint: 3 questions, 8 marks, pass 80%
Practise step 2
Step 3

Full marks on long answers

You can expand, evaluate and factorise in multi-part questions and use identities for areas and volumes.

Read first: 3. Cube identities; 4. Factorising; Worked examples 1-3 3 practice questions · checkpoint: 2 questions, 9 marks, pass 80%
Practise step 3
Step 4

95+ stretch (HOTS)

You can chain identities, such as from x − 1/x to x⁴ + 1/x⁴, and set up identity-based models.

Read first: 2. Rearrangements; 3. Cube identities; Common errors 3 practice questions · checkpoint: 3 questions, 9 marks, pass 80%
Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 3 of the 17 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceMental arithmetic with identities

Using an identity, \(103^2\) equals

  1. (a)\(10900\)
  2. (b)\(10069\)
  3. (c)\(10609\)
  4. (d)\(10309\)
Q2·1 mark·Multiple choiceFactorising with identities

\(49x^2 - 25\) factorises as

  1. (a)\((7x - 5)(7x + 5)\)
  2. (b)\((7x - 5)^2\)
  3. (c)\((49x - 5)(x + 5)\)
  4. (d)\((7x + 5)^2\)
Q3·1 mark·Multiple choiceSquare identities

If \(a + b = 7\) and \(ab = 10\), then \(a^2 + b^2\) is

  1. (a)\(49\)
  2. (b)\(29\)
  3. (c)\(39\)
  4. (d)\(69\)

Stretch yourself: original algebra problems at Intermediate level (ages 13 to 16), with hints and full solutions: Problem I24 · Problem I106 · Problem I113 · Problem I120. All 29 →

Where marks are lost in Exploring Algebraic Identities

  • Missing middle term: (a + b)² written as a² + b². Fix: always write the 2ab term.
  • Coefficients not squared: (3x − 2y)² with 3x² instead of 9x². Fix: bracket each term before squaring.
  • Sign errors in (a + b − c)². Fix: treat it as (a + b + (−c))² and carry the minus sign into 2bc and 2ca.
  • Wrong bracket in sums/differences of cubes. Fix: a³ − b³ = (a − b)(a² + ab + b²): the sign in (a − b) is the same as in a³ − b³, the middle term ab takes the opposite sign, and the last term is always plus.
  • x³ + 1/x³ found as (x + 1/x)³. Fix: subtract 3(x + 1/x).

Test Exploring Algebraic Identities against the clock

Want it against the clock? Take a timed 30-mark chapter test on Exploring Algebraic Identities, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Essentials or the free trial).