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CBSE · Class 9 · Mathematics

Chapter 4: Algebraic Identities

5 concepts · 40 practice questions with answers · Free to read

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Squares of sums and differences

From daily life

A square garden of side (a + b) metres is split into a square of side a, a square of side b and two a × b paths. Adding the four areas gives its total area.

Simple definition

An identity is an equation true for every value of the letters. The two basic ones expand the square of a binomial without long multiplication.

Formula: (a + b)² = a² + 2ab + b² (a − b)² = a² − 2ab + b²

a²ababb²

In the figure: The big square has area (a + b)², and also a² + ab + ab + b².

Rule 1: Expand a sum square — Square the first, double the product, square the second.

  • Easy: (x + 5)² = x² + 10x + 25.
  • Medium: (2m + 3)² = 4m² + 12m + 9.
  • Exam-type: Mental: 103² = (100 + 3)² = 10000 + 600 + 9 = 10609.

Rule 2: Expand a difference square — Same pattern, but the middle term is negative.

  • Easy: (y − 4)² = y² − 8y + 16.
  • Medium: (3p − 2q)² = 9p² − 12pq + 4q².
  • Exam-type: Mental: 98² = (100 − 2)² = 10000 − 400 + 4 = 9604.
Remember

Both squares have the same first and last terms; only the sign of 2ab changes. (a + b)² and (−a − b)² are equal.

Common mistake

Writing (a + b)² = a² + b², which leaves out 2ab.

Memory clue

"First square, twice the cross, last square."

Quick table

IdentityMiddle term
(a + b)²+ 2ab
(a − b)²− 2ab

Practice questions

Easy
  1. 1Expand (z + 2)².
  2. 2Expand (z − 4)².
  3. 3Find 99² using an identity.
Medium
  1. 4Expand (2x + 3)².
  2. 5If x + y = 6 and xy = 7, find x² + y².
  3. 6What number must be added to x² + 18x to make it a perfect square?
Challenging
  1. 7If x + 1/x = 4, find x² + 1/x².
  2. 8If x − y = 2 and x² + y² = 100, find xy.
Show answers (try them yourself first)
  1. 1z² + 4z + 4
  2. 2z² − 8z + 16
  3. 39801
  4. 44x² + 12x + 9
  5. 522
  6. 681
  7. 714
  8. 848
Live figure

Squares of sums and differences

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Difference of squares and (a + b + c)²

From daily life

Cut a small b × b square out of a big a × a sheet and rearrange the rest into a rectangle: its sides are (a + b) and (a − b).

Simple definition

A difference of two squares factors as a product of the sum and the difference. For three terms, the square has three squares and three double products.

Formula: a² − b² = (a + b)(a − b) (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

(a−b)²b(a−b)b(a−b)b²

In the figure: Remove the red b²: what is left, (a − b)² + 2b(a − b), equals (a − b)(a + b).

Rule 1: Difference of squares — Write each term as a square, then use sum × difference.

  • Easy: 49 − x² = (7 + x)(7 − x).
  • Medium: Mental: 101 × 99 = 100² − 1² = 9999.
  • Exam-type: 25a² − 16b² = (5a + 4b)(5a − 4b).

Rule 2: Square of three terms — Square every term, then add twice each pair product.

  • Easy: (x + y + 1)² = x² + y² + 1 + 2xy + 2y + 2x.
  • Medium: (a − b + c)² = a² + b² + c² − 2ab − 2bc + 2ca; check the sign of each pair.
  • Exam-type: If a + b + c = 0 then a² + b² + c² = −2(ab + bc + ca).
Remember

Only a difference (minus sign) of two perfect squares factors this way. A sum of squares a² + b² does not factor over the real numbers.

Common mistake

Writing a² − b² = (a − b)².

Memory clue

"Square minus square = sum times difference."

Quick table

PatternFactors
x² − 9(x + 3)(x − 3)
4y² − 1(2y + 1)(2y − 1)
a² − 25b²(a + 5b)(a − 5b)

Practice questions

Easy
  1. 1Expand (x + 8)(x − 8).
  2. 2Evaluate 52² − 44² using an identity.
  3. 3Find 29 × 31 using an identity.
Medium
  1. 4Find 94 × 106 using an identity.
  2. 5In the expansion of (x + 3y + 2z)², what is the coefficient of yz?
  3. 6If a + b + c = 9 and ab + bc + ca = 3, find a² + b² + c².
Challenging
  1. 7If x + y = 3 and x − y = 6, find x² − y².
  2. 8Expand (x + 1y + 4)².
Show answers (try them yourself first)
  1. 1x² − 64
  2. 2768
  3. 3899
  4. 49964
  5. 512
  6. 675
  7. 718
  8. 8x² + y² + 16 + 2xy + 8y + 8x
Live figure

Difference of squares and (a + b + c)²

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The identity (x + a)(x + b)

From daily life

A rectangular room measures (x + 2) m by (x + 3) m. Its floor area is the x² square plus two strips and a small 2 × 3 corner.

Simple definition

When two binomials share the same first term x, the product has x², then the sum of the numbers times x, then the product of the numbers.

Formula: (x + a)(x + b) = x² + (a + b)x + ab

x²3x2x6

In the figure: The four parts x², 3x, 2x and 6 add up to x² + 5x + 6.

Rule 1: Multiply two binomials — Add the numbers for the middle coefficient, multiply them for the last term. Keep the signs.

  • Easy: (x + 4)(x + 7) = x² + 11x + 28.
  • Medium: (y − 3)(y + 5) = y² + 2y − 15.
  • Exam-type: Mental: 102 × 105 = (100 + 2)(100 + 5) = 10000 + 700 + 10 = 10710.

Rule 2: Special cases — If a = b you get a square; if a = −b you get a difference of squares.

  • Easy: (x + 3)(x + 3) = x² + 6x + 9.
  • Medium: (x + 6)(x − 6) = x² − 36.
  • Exam-type: If x² + 9x + 20 = (x + a)(x + b) then a + b = 9 and ab = 20, so a, b = 4, 5.
Remember

The middle coefficient is the sum of the numbers; the constant term is their product, both with signs.

Common mistake

Writing the middle term as ab instead of a + b.

Memory clue

"Sum in the middle, product at the end."

Quick table

(x + a)(x + b)Middle / End
(x + 2)(x + 3)5x / 6
(x − 1)(x + 4)3x / −4
(x − 2)(x − 5)−7x / 10

Practice questions

Easy
  1. 1Expand (x + 4)(x + 3).
  2. 2In the expansion of (x + 7)(x + 8), what is the coefficient of x?
  3. 3Expand (x − 2)(x − 3).
Medium
  1. 4Find 33 × 36 by writing it as (x + a)(x + b) with x = 30.
  2. 5Expand (x + 8)(x − 5).
  3. 6x² + 11x + 24 equals (x + p)(x + q). What is p + q?
Challenging
  1. 7For what value of k is (x + 2)(x + k) = x² + 9x + 14?
  2. 8Simplify (x + 6)(x + 6) − (x + 5)(x + 4).
Show answers (try them yourself first)
  1. 1x² + 7x + 12
  2. 215
  3. 3x² − 5x + 6
  4. 41188
  5. 5x² + 3x − 40
  6. 611
  7. 77
  8. 83x + 16

Factorisation using identities

From daily life

Factorising is like finding the dimensions of a rectangle when only its area is known. Identities tell us which dimensions to try.

Simple definition

Factorising writes an expression as a product. First take out the common factor, then look for an identity: perfect square, difference of squares, or x² + (a + b)x + ab.

Formula: a² + 2ab + b² = (a + b)² a² − 2ab + b² = (a − b)² a² − b² = (a + b)(a − b)

Common factor→Spot identity→Check

In the figure: Always take out the common factor first, then decide which identity fits, then multiply back to check.

Rule 1: Perfect squares — If the first and last terms are squares and the middle is twice their product, it is a perfect square.

  • Easy: x² + 12x + 36 = (x + 6)².
  • Medium: 9a² − 12a + 4 = (3a − 2)².
  • Exam-type: Common factor first: 2x² + 16x + 32 = 2(x + 4)².

Rule 2: Trinomials x² + px + q — Find two numbers with sum p and product q.

  • Easy: x² + 7x + 12 = (x + 3)(x + 4).
  • Medium: x² − 5x + 6 = (x − 2)(x − 3).
  • Exam-type: x² + x − 12 = (x + 4)(x − 3).
Remember

Multiply the factors back; you must get the original expression.

Common mistake

Stopping early: 3x² − 12 = 3(x² − 4) is not finished; it is 3(x + 2)(x − 2).

Memory clue

"Common factor, identity, check."

Quick table

ExpressionFactors
x² + 10x + 25(x + 5)²
x² − 49(x + 7)(x − 7)
x² + 8x + 15(x + 3)(x + 5)

Practice questions

Easy
  1. 1Factorise 35x + 7.
  2. 2Factorise y² − 144.
  3. 3Factorise x² − 11x + 18.
Medium
  1. 4Factorise 4x² − 36.
  2. 5Factorise 9x² + 12x + 4.
  3. 6Factorise completely: 2x² + 8x + 6.
Challenging
  1. 7If x² + px + 36 has factors (x + 6) and (x + 6), find p.
  2. 8Factorise x⁴ − 1.
Show answers (try them yourself first)
  1. 17(5x + 1)
  2. 2(y + 12)(y − 12)
  3. 3(x − 2)(x − 9)
  4. 4(2x + 6)(2x − 6)
  5. 5(3x + 2)²
  6. 62(x + 3)(x + 1)
  7. 712
  8. 8(x² + 1)(x + 1)(x − 1)

Simplifying rational expressions

From daily life

A fraction 12/18 becomes 2/3 after cancelling the common factor 6. Algebraic fractions are simplified the same way, but only common factors can be cancelled.

Simple definition

A rational expression is a fraction of two polynomials. To simplify, factorise the top and the bottom, then cancel common factors. The bottom can never be zero.

Formula: (x² − a²)/(x − a) = x + a for x ≠ a

(x²−4)/(x+2)→(x+2)(x−2)/(x+2)→x − 2

In the figure: Factorise the top, cancel (x + 2), and what remains is x − 2.

Rule 1: Cancel factors, not terms — Only a factor that multiplies the whole top and the whole bottom can be cancelled.

  • Easy: (5x + 10)/5 = (5(x + 2))/5 = x + 2.
  • Medium: (x² − 9)/(x + 3) = x − 3, for x ≠ −3.
  • Exam-type: (x + 3)/3 cannot be cut to x + 1; the 3 is not a factor of x + 3.

Rule 2: Multiply and divide fractions — Factorise everything first, then cancel across the fractions.

  • Easy: (x²/(x − 1)) × ((x − 1)/x) = x.
  • Medium: Dividing by a fraction means multiplying by its reciprocal.
  • Exam-type: Check the answer by putting a value of x, such as x = 3.
Remember

Always say which values of x are not allowed: those that make the bottom zero.

Common mistake

Cancelling a term: (x + 4)/4 ≠ x + 1.

Memory clue

"Factorise first, then cancel."

Quick table

BeforeAfter
(x² − 1)/(x − 1)x + 1
(2x + 6)/2x + 3
(x² + 5x + 6)/(x + 2)x + 3

Practice questions

Easy
  1. 1Find the value of (3x + 5)/(x + 1) at x = 3.
  2. 2Simplify 20x/4x (x ≠ 0).
  3. 3Simplify (x + 3)(x + 7)/(x + 3), where x + 3 ≠ 0.
Medium
  1. 4Simplify (x² − 4)/(x + 2), where x ≠ −2.
  2. 5Simplify (x² − 25)/(x + 5), where x ≠ −5.
  3. 6Simplify (4x + 20)/(16x + 80).
Challenging
  1. 7Simplify (3x + 9)/(9x + 27).
  2. 8Simplify [(x² − 16)/(x + 7)] × [(x + 7)/(x − 4)].
Show answers (try them yourself first)
  1. 17/2
  2. 25
  3. 3x + 7
  4. 4x − 2
  5. 5x − 5
  6. 61/4
  7. 71/3
  8. 8x + 4