6.6 Going Deeper: Enrichment & Exam Preparation

Where Identities Show Up in Real Life

Math in the Real World

  • Mental-maths shortcuts. A shopkeeper pricing 103 items at ₹103 each computes 103^2 = (100+3)^2 = 10000 + 600 + 9 = 10609 in one breath, no calculator. The same trick squares any “almost-round” number.
  • Quick products at the till. To find 98 \times 102, notice the numbers straddle 100: (100-2)(100+2) = 100^2 - 2^2 = 9996. Pairs that share a midpoint are a difference of squares in disguise.
  • Areas and borders. A square plot of side a+b splits into a field a^2, a second field b^2, and two identical ab access strips, exactly the picture behind (a+b)^2 = a^2 + 2ab + b^2. Carpenters, tilers and farmers use this every day to estimate material.
  • Number tricks and magic. The “think of a number, square it, subtract the square of the next-door number” puzzle works because (n+1)^2 - n^2 = 2n+1, always the next odd number. Identities turn party tricks into provable facts.
  • A bridge to later chapters. Factorising with a^2 - b^2 = (a+b)(a-b) is the first step into algebraic factorisation and solving equations, and (a\pm b)^2 feeds directly into completing the square.

Exam Tip The single most common slip in this chapter is dropping the middle term: writing (a+b)^2 = a^2 + b^2. Always remember the 2ab. A two-second sanity check is to put a=b=1: the true value is (1+1)^2 = 4, while a^2+b^2 = 2. If your expanded formula does not give 4 here, the middle term is missing.

Quick Reference: Identities & Fast-Multiply Shortcuts

Memorising these turns long multiplications into one-line mental sums, a real time-saver under exam pressure.

Identity Expansion Used to …
(a+b)^2 a^2 + 2ab + b^2 square a sum, e.g. 103^2
(a-b)^2 a^2 - 2ab + b^2 square a difference, e.g. 98^2
(a+b)(a-b) a^2 - b^2 products straddling a centre, e.g. 98\times102
(a+m)(b+n) ab + mb + an + mn any two-bracket product (FOIL)
(a+b)^2+(a-b)^2 2(a^2+b^2) sum/difference identities together
(a+b)^2-(a-b)^2 4ab recover ab from the two squares
Fast multiply Split as Example
\times 11 \times(10+1) 72\times11 = 720+72 = 792
\times 101 \times(100+1) 76\times101 = 7600+76 = 7676
\times 99 \times(100-1) 47\times99 = 4700-47 = 4653
\times 999 \times(1000-1) 58\times999 = 58000-58 = 57942
ends-in-5 square (10t+5)^2 = 100\,t(t{+}1)+25 85^2:\ 8\times9=72 \to 7225
near a round N (N{-}d)(N{+}d) = N^2 - d^2 97\times103 = 10000-9 = 9991

Use the explorer to expand or factorise any (a\pm b)^2 or a^2-b^2 and watch the identity verified numerically:

Identity expand & verify

Memory Tricks & One-Page Revision

Quick Revision Card

  • Distributive law: a(b+c) = ab + ac, multiply by every term inside the bracket.
  • The three identities: (a+b)^2 = a^2 + 2ab + b^2; \ (a-b)^2 = a^2 - 2ab + b^2; \ (a+b)(a-b) = a^2 - b^2.
  • Never drop the 2ab. (a+b)^2 \neq a^2 + b^2.
  • Two-bracket FOIL: (a+m)(b+n) = ab + mb + an + mn, first \times first, then cross terms, then last \times last.
  • Square anything near a round number: (N\pm d)^2 = N^2 \pm 2Nd + d^2. (E.g. 103^2 = 10000 + 600 + 9.)
  • Product straddling a centre N: (N-d)(N+d) = N^2 - d^2. (E.g. 98\times102 = 9996.)
  • Ends-in-5 square: front part \times next number, then write 25. (75^2:\ 7\times8 = 56 \to 5625.)
  • Two handy combinations: (a+b)^2 + (a-b)^2 = 2(a^2+b^2) and (a+b)^2 - (a-b)^2 = 4ab.

Spot the Mistake

Common Exam Mistakes

  • Writing (a+b)^2 = a^2 + b^2, the middle term 2ab is essential, so (a+b)^2 = a^2 + 2ab + b^2.
  • Flipping the wrong sign in (a-b)^2: only the middle term turns negative, a^2 - 2ab + b^2, the +b^2 stays positive since (-b)^2 = b^2.
  • Treating (a+b)(a-b) as a^2 + b^2. It is a^2 - b^2; the cross terms -ab and +ab cancel.
  • Forgetting to square the coefficient: (5m)^2 = 25m^2, not 5m^2. So (5m+6n)^2 = 25m^2 + 60mn + 36n^2.
  • Combining unlike terms: 5w^2 + 6w is already simplified, different powers of w cannot be added.
  • Mis-distributing a negative: -3p(-5p+2q) = 15p^2 - 6pq, not -3p + 5p - 2q.

Train Your Mental Multiplication

A quick drill: an identity-friendly product or square appears, and you type the answer. The hint shows which identity to lean on. Build a streak!

Fast-multiply trainer
Use an identity, then type the value and press Check.
 
Score 0 · Streak 0

Exam Corner: CBSE-Style Practice

Mixed Practice (objective, short, long, HOTS)

Objective type (1 mark each)

  1. The expansion of (a-b)^2 is ______.
  2. Using an identity, $103^2 = $ ______.
  3. The value of 98 \times 102 is ______.

Short answer (2 marks each)

  1. Expand (2x+3)^2 and (3a-5b)^2.
  2. Evaluate 204^2 using a suitable identity, showing each step.

Long answer (3 marks each)

  1. Evaluate 83^2 - 17^2 without squaring either number, by recognising a difference of squares.
  2. Factorise 9x^2 - 25 and x^2 - 14x + 49, naming the identity used in each case.

HOTS (Higher Order Thinking)

  1. If a + b = 10 and ab = 21, find a^2 + b^2 without finding a and b separately.
  2. Show that for any number n, the value of (n+1)^2 - n^2 is always the next odd number, and use this to write 51^2 - 49^2 as a single product.

Assertion–Reason (Choose: (a) both true, R explains A; (b) both true, R does not explain A; (c) A true, R false; (d) A false, R true.)

  1. Assertion (A): 35 \times 45 = 1575.   Reason (R): Two numbers symmetric about a centre N satisfy (N-d)(N+d) = N^2 - d^2.
  1. (a-b)^2 = \mathbf{a^2 - 2ab + b^2}.
  2. 103^2 = (100+3)^2 = 10000 + 600 + 9 = \mathbf{10609}.
  3. 98 \times 102 = (100-2)(100+2) = 10000 - 4 = \mathbf{9996}.
  4. (2x+3)^2 = (2x)^2 + 2(2x)(3) + 3^2 = \mathbf{4x^2 + 12x + 9};   (3a-5b)^2 = (3a)^2 - 2(3a)(5b) + (5b)^2 = \mathbf{9a^2 - 30ab + 25b^2}.
  5. 204^2 = (200+4)^2 = 200^2 + 2(200)(4) + 4^2 = 40000 + 1600 + 16 = \mathbf{41616}.
  6. 83^2 - 17^2 = (83+17)(83-17) = 100 \times 66 = \mathbf{6600}.
  7. 9x^2 - 25 = (3x)^2 - 5^2 = \mathbf{(3x+5)(3x-5)}, difference of squares, Identity 1C.   x^2 - 14x + 49 = x^2 - 2(x)(7) + 7^2 = \mathbf{(x-7)^2}, square of a difference, Identity 1B.
  8. (a+b)^2 = a^2 + 2ab + b^2, so a^2 + b^2 = (a+b)^2 - 2ab = 10^2 - 2(21) = 100 - 42 = \mathbf{58}.
  9. (n+1)^2 - n^2 = (n^2 + 2n + 1) - n^2 = 2n + 1, which is the next odd number after 2n. For the second part, 51^2 - 49^2 = (51+49)(51-49) = 100 \times 2 = \mathbf{200}.
  10. (a), Both true and R explains A: 35 \times 45 = (40-5)(40+5) = 40^2 - 5^2 = 1600 - 25 = 1575, exactly the difference-of-squares pattern.

Connections to Other Chapters

How This Chapter Links Forward

  • Chapter 1 (A Square and A Cube): the ends-in-5 squaring trick and (a+b)^2 expansion are the same machinery you met for squares; here we prove why they work.
  • Chapter 13 (Algebra Play): running the identities backwards, a^2 - b^2 \to (a+b)(a-b), a^2 + 2ab + b^2 \to (a+b)^2, is the heart of factorisation and solving equations.
  • Chapter 9 (Baudhayana–Pythagoras Theorem): rearranging four right triangles inside a square uses (a+b)^2 = a^2 + 2ab + b^2 to prove a^2 + b^2 = c^2.
  • Chapter 14 (Area): every identity in this chapter is, at heart, a statement about how a rectangle’s area splits into smaller pieces.

Glossary

Key Terms

  • Distributive property: the rule a(b+c) = ab + ac; the foundation of all multiplication “by parts.”
  • Identity: an equation true for every value of its letters, such as (a+b)^2 = a^2 + 2ab + b^2.
  • Expansion: rewriting a product of brackets as a sum of terms, e.g. (x+2)(x+5) = x^2 + 7x + 10.
  • Factorisation: the reverse of expansion, writing a sum as a product, e.g. a^2 - b^2 = (a+b)(a-b).
  • Like terms: terms with identical letters to identical powers; only these may be added or subtracted.
  • Difference of two squares: the expression a^2 - b^2, which always factorises as (a+b)(a-b).
  • Perfect-square trinomial: an expression of the form a^2 \pm 2ab + b^2, equal to (a\pm b)^2.