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You will learn: Use identities to simplify calculations and turn expressions into useful products.

Before you start: Multiplying integers and simple brackets; letters represent real numbers unless stated otherwise.

An identity works for every allowed value

An equation such as x + 2 = 5 asks which x works. An identity, such as (a + b)² = a² + 2ab + b², is true for all real a and b. Identities let us choose a useful shape for an expression.

Expand one bracket at a time: (a + b)(a + b) = a² + ab + ba + b². The two middle products are both ab, so their sum is 2ab. This is why the square of a sum includes a middle term.

A square split into four rectanglesFor positive lengths a and b, a square of side a plus b is split into areas a squared, ab, ab and b squared.a²ababb²abab
The area picture explains the identity for positive lengths. The algebra proves it for all real values.

The difference of two squares

Multiplying (a − b)(a + b) gives a² + ab − ab − b² = a² − b². The middle terms cancel. Thus a² − b² = (a − b)(a + b). This factors a difference of squares; it does not say that a² + b² is (a + b)².

Worked example 1: a quick calculation

Calculate 101² − 99². Instead of squaring both numbers, factor the difference: (101 − 99)(101 + 99) = 2 × 200 = 400. The small difference and round sum make this form useful.

Worked example 2: use a known sum and product

Suppose x + y = 10 and xy = 21. Find x² + y². From (x + y)² = x² + 2xy + y², rearrange to x² + y² = (x + y)² − 2xy. Substitute the given values: 100 − 42 = 58. There is no need to find x and y separately.

A first inequality

A real square cannot be negative, so (x − y)² ≥ 0. Expanding gives x² + y² ≥ 2xy. Equality holds exactly when x − y = 0, or x = y. The equality case is part of the result, not an optional detail.

Check the middle term

The tempting claim (a + b)² = a² + b² fails at a = b = 1: it would say 4 = 2. A small test can disprove a false identity. To establish a true identity for all values, use expansion or another general argument.

Practise at your next step

A six-question session chooses from nine questions. Two correct answers in a row without hints move you up a level; an incorrect answer brings a simpler next question where one is available. Hints keep you at the same level. This is a practice suggestion, not an exam score or proof of mastery.

Interactive practice loads here. You can also use the complete question set below.

Write a proof of your own

If x and y are real numbers with x + y = 12, prove xy ≤ 36 and state when equality holds.

Write your reasoning on paper before comparing. The practice checker does not grade a written proof.

Compare your proof with a full solution

Since (x − y)² ≥ 0, we have x² + y² ≥ 2xy. Adding 2xy gives (x + y)² ≥ 4xy. Substitute x + y = 12 to obtain 144 ≥ 4xy, so xy ≤ 36. Equality requires x = y; combined with their sum, this gives x = y = 6, which does attain 36.

Check: did you state the assumptions, explain the key step, and reach the requested conclusion?

All nine practice questions, hints and solutions

This complete set works without the interactive practice. Hide each solution until you have made an attempt.

1. Which is the correct expansion of (x + 3)²?

Foundation

  1. x² + 9
  2. x² + 3x + 9
  3. x² + 6x + 9
  4. 2x + 6
Hint

Multiply (x + 3) by another (x + 3).

Answer and explanation

x² + 6x + 9. The four products are x², 3x, 3x and 9. Adding them gives x² + 6x + 9.

2. Which product equals x² − 16?

Foundation

  1. (x − 4)²
  2. (x − 4)(x + 4)
  3. (x − 8)(x + 2)
  4. (x + 4)²
Hint

16 is the square of 4.

Answer and explanation

(x − 4)(x + 4). Use a² − b² = (a − b)(a + b) with a = x and b = 4. This gives x² − 16 = (x − 4)(x + 4).

3. What is 51² − 49²?

Foundation

  1. 2
  2. 100
  3. 200
  4. 2500
Hint

Factor before multiplying.

Answer and explanation

200. 51² − 49² = (51 − 49)(51 + 49) = 2 × 100 = 200.

4. If x + y = 9 and xy = 14, what is x² + y²?

Core

  1. 25
  2. 53
  3. 67
  4. 81
Hint

Rearrange the identity for the square of a sum.

Answer and explanation

53. x² + y² = (x + y)² − 2xy = 9² − 2 × 14 = 81 − 28 = 53.

5. For real x, what is the smallest possible value of (x − 4)² + 7?

Core

  1. 0
  2. 4
  3. 7
  4. 11
Hint

A square is nonnegative, and can equal zero.

Answer and explanation

7. The square is at least 0, so the expression is at least 7. At x = 4 the square is 0, and the value 7 is attained. Therefore 7 is the minimum.

6. Which expression equals (n + 1)² − n²?

Core

  1. 1
  2. 2n + 1
  3. n + 1
  4. 2n² + 1
Hint

Expand (n + 1)² and subtract n².

Answer and explanation

2n + 1. (n + 1)² − n² = n² + 2n + 1 − n² = 2n + 1. The two n² terms cancel.

7. If real x and y satisfy x + y = 8, what is the largest possible value of xy?

Stretch

  1. 8
  2. 12
  3. 16
  4. 32
Hint

Use (x − y)² ≥ 0 and the known sum.

Answer and explanation

16. (x − y)² ≥ 0 gives (x + y)² ≥ 4xy. Thus 64 ≥ 4xy and xy ≤ 16. The bound is attained at x = y = 4, so 16 is the maximum.

8. For an integer n, what is (n + 2)² − (n − 2)²?

Stretch

  1. 4n
  2. 8n
  3. 8
  4. 4n²
Hint

Use the difference-of-squares identity.

Answer and explanation

8n. The difference is [(n + 2) − (n − 2)][(n + 2) + (n − 2)] = 4 × 2n = 8n. In fact, this identity also holds for real n.

9. Positive real numbers a and b satisfy a/b + b/a = 2. What follows?

Stretch

  1. a = b
  2. a + b = 2
  3. ab = 1
  4. a = 2b
Hint

Multiply by the positive number ab, then collect the terms into a square.

Answer and explanation

a = b. Multiplying gives a² + b² = 2ab, so (a − b)² = 0. A real square equals 0 only when its base is 0. Hence a = b.

Where to go next

Move on to the mixed proof-practice plan · Explore the wider Olympiad topic map

Teaching examples prepared for this guide, not attributed to a contest paper. Familiar elementary problems may appear in other learning materials. Report an unclear step or an error.