Class 9 Maths · Original EZT practice
Exploring Algebraic Identities (Class 9 Chapter 4): summary and practice
An identity holds for every value of its letters; an equation may hold only for some. Chapter 4 turns that idea into fast expansion, quick mental squares and clean factorisation. All questions are original EZT tutor practice following the chapter's own section order. Edition note: this page covers the new NCERT book Ganita Manjari Part I (Grade 9), First Edition April 2026. Older Class 9 books numbered chapters differently - check your chapter title, not just the number.
8 original worked examples, 8 fresh self-check questions and 4 mixed questions, with explained answers. These are not copied textbook exercises, an official chapter index or a full answer key. No affiliation with NCERT is claimed.
Official sources for topic and edition context, checked 20 October 2026:
1. Identity versus equation
An identity is true for every value of its variables; an equation can be true only for some. The square of a sum always equals the sum of the squares plus twice the product - that is what makes it an identity.
2. The core identities
Square of a sum: (a + b)^2 = a^2 + 2ab + b^2. Square of a difference: (a - b)^2 = a^2 - 2ab + b^2. Square of a three-term sum: (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca. Difference of squares: a^2 - b^2 = (a + b)(a - b).
3. Factorisation using identities
Read an expression backwards through an identity: a perfect-square trinomial collapses into a square, and a difference of squares splits into two brackets.
4. Simplifying rational expressions
Factor the numerator and denominator first, then cancel any common factor. Identities do most of the heavy lifting.
Choose, solve, check, then try a fresh question.
Before calculating, read the method prompt and name the condition that makes it useful. Write your own attempt. Open the steps only when you need them, and use the check to catch a wrong sign, unit or interpretation. Finish with the fresh question before reading its answer.
Tell an identity from an equation
Choose the method: An identity must hold for every value; test another value before deciding.
EZT practice question
Is x^2 - 1 = 24 an identity?
Show the worked solution
- Try x = 5: 25 - 1 = 24, true.
- Try x = 1: 1 - 1 = 0, which is not 24.
Answer: No - it holds only for x = 5 and x = -5, so it is an equation, not an identity.
Check: An identity must hold for every value of the variable.
Watch for this: Do not call a statement an identity after checking a single value.
Now try it yourself
Verify (a + b)^2 = a^2 + 2ab + b^2 for a = -2/3 and b = 3/4.
Check your answer
Both sides equal 1/144.
a + b is 1/12, and squaring gives 1/144; the right side adds up to the same.
Prove a pattern with consecutive squares
Choose the method: Expand both squares and watch the variable terms cancel.
EZT practice question
Show that (n - 1)^2 + (n + 1)^2 - 2n^2 has the same value for every n.
Show the worked solution
- (n - 1)^2 = n^2 - 2n + 1 and (n + 1)^2 = n^2 + 2n + 1.
- Adding them gives 2n^2 + 2.
- Subtracting 2n^2 leaves 2.
Answer: It equals 2 for every n.
Check: With n = 5: 16 + 36 - 50 = 2.
Watch for this: Do not test one value and stop; only the expansion proves it for all n.
Now try it yourself
Check the pattern for n = 7.
Check your answer
36 + 64 - 98 = 2.
(7 - 1)^2 = 36 and (7 + 1)^2 = 64, and twice 49 is 98.
Square a two-digit number mentally
Choose the method: Split the number into a round part and a small part, then use the square of a sum.
EZT practice question
Find 43 squared using an identity.
Show the worked solution
- Write 43 as 40 + 3.
- (40 + 3)^2 = 1600 + 240 + 9.
Answer: 1849.
Check: 43 x 43 = 1849 by direct multiplication.
Watch for this: Do not write 1600 + 9; the middle term 2ab is 240.
Now try it yourself
Find 29 squared the same way.
Check your answer
841.
(30 - 1)^2 = 900 - 60 + 1.
Square a three-digit number with the three-term identity
Choose the method: Split into three parts and include all three cross terms.
EZT practice question
Find 119 squared using (a + b + c)^2.
Show the worked solution
- Write 119 as 100 + 10 + 9.
- Squares: 10000 + 100 + 81; cross terms: 2000 + 1800 + 180.
- Add everything.
Answer: 14161.
Check: 119 x 119 = 14161 directly.
Watch for this: There are three cross terms, not two; missing one breaks the total.
Now try it yourself
Find 105 squared.
Check your answer
11025.
(100 + 5)^2 = 10000 + 1000 + 25.
Use the difference of squares for fast squaring
Choose the method: Write a^2 as (a + b)(a - b) + b^2 with a convenient b.
EZT practice question
Find 55 squared using a^2 = (a + b)(a - b) + b^2.
Show the worked solution
- Choose b = 5: (55 + 5)(55 - 5) + 25.
- Sixty times fifty is 3000.
- Add 25.
Answer: 3025.
Check: 55 x 55 = 3025 directly.
Watch for this: Do not forget to add b^2 back at the end.
Now try it yourself
Find 98 squared with the same trick.
Check your answer
9604.
(98 + 2)(98 - 2) + 4 = 9600 + 4.
Expand the square of a binomial
Choose the method: Apply (a + b)^2 = a^2 + 2ab + b^2 with the full terms.
EZT practice question
Expand (5x + 2y)^2.
Show the worked solution
- Take a = 5x and b = 2y.
- a^2 = 25x^2, 2ab = 20xy and b^2 = 4y^2.
Answer: 25x^2 + 20xy + 4y^2.
Check: The middle term is 2 x 5x x 2y = 20xy.
Watch for this: Do not write 25x^2 + 4y^2; the middle term does not vanish.
Now try it yourself
Expand (3x - 4)^2.
Check your answer
9x^2 - 24x + 16.
The middle term carries the minus sign: 2 x 3x x (-4) = -24x.
Recognise a perfect-square trinomial
Choose the method: Check that the two end terms are squares and the middle term matches 2ab.
EZT practice question
Factorise 49x^2 + 14x + 1.
Show the worked solution
- 49x^2 is (7x)^2 and 1 is 1^2.
- The middle term is 2 x 7x x 1 = 14x, which matches.
Answer: (7x + 1)^2.
Check: Expanding (7x + 1)^2 returns 49x^2 + 14x + 1.
Watch for this: Check the middle term before claiming a perfect square.
Now try it yourself
Factorise 4x^2 - 12x + 9.
Check your answer
(2x - 3)^2.
The middle term is 2 x 2x x (-3) = -12x.
Factorise a trinomial by splitting the middle term
Choose the method: Find two numbers whose sum is the middle coefficient and whose product is the constant.
EZT practice question
Factorise x^2 + 9x + 20.
Show the worked solution
- Find two numbers adding to 9 with product 20: four and five.
- So x^2 + 9x + 20 = (x + 4)(x + 5).
Answer: (x + 4)(x + 5).
Check: Four plus five is nine, and four times five is twenty.
Watch for this: Watch the signs: for x^2 - 7x + 12 the pair is -3 and -4.
Now try it yourself
Factorise x^2 + 8x + 15.
Check your answer
(x + 3)(x + 5).
Three plus five is eight, and three times five is fifteen.
Try a mixed set
Choose a method or explanation before opening each answer.
1. State the square-of-a-sum identity.
Show answer and reason
(a + b)^2 = a^2 + 2ab + b^2.
It holds for every value of a and b.
2. What is the difference of squares?
Show answer and reason
a^2 - b^2 = (a + b)(a - b).
It turns a subtraction into two brackets.
3. How is an identity different from an equation?
Show answer and reason
An identity holds for all values; an equation may hold only for some.
Checking one value can never prove an identity.
4. How do you factor x^2 + bx + c quickly?
Show answer and reason
Find two numbers with sum b and product c.
They become the constants in the two brackets.
Try it yourself
Work these on paper before checking any notes. Answers are deliberately not shown here.
- Find 103 squared using an identity.
- Factorise 9x^2 + 6x + 1.
- Is (a + b - c)^2 + (a - b + c)^2 + (a - b - c)^2 equal to 2a^2 + 2b^2 + 2c^2? Test with a = b = c = 1 before deciding.
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Questions about this practice page
- What is the difference between an identity and an equation?
- An identity is true for every value of its variables; an equation may be true only for particular values.
- Can I use (a + b)^2 with fractions or negative numbers?
- Yes - identities hold for all values, including fractions and negatives.
- How do I factorise x^2 + bx + c quickly?
- Find two numbers whose sum is b and whose product is c; they complete the two brackets.
- Why does (n - 1)^2 + (n + 1)^2 - 2n^2 always give 2?
- Expanding both squares leaves 2n^2 + 2, and subtracting 2n^2 leaves 2 for every n.
- Are these the NCERT exercise questions for Exploring Algebraic Identities (Class 9)?
- No. EZT pages are original practice following the chapter's section order - use them to prepare for the textbook exercises, not as a substitute for your book.