Difference of squares

The idea

(a + b)(a – b) = a² – b²: the product of the sum and the difference of two numbers equals the difference of their squares.

A pattern of squares

9 × 9 − 1 × 1 = 10 × 8 8 × 8 − 6 × 6 = 14 × 2 7 × 7 − 2 × 2 = 9 × 5 10 × 10 − 4 × 4 = 14 × 6

Spot the pattern. What goes in the boxes? 6 × 6 − 4 × 4 = □ × □

• 10 × 2
• 2 × 2
• 24 × 2

Cut, turn and slide

Each picture is a square with a b × b corner missing. Cut the L into two rectangles. Turn the small piece and slide it. What shape do you get?

A square minus a square is a rectangle

(a + b)(a − b) = a² − b²: the product of the sum and the difference of two numbers equals the difference of their squares.

(a + b)(a − b) = a² − ab + ba − b² = a² − b² The −ab and +ba cancel. NCERT calls this Identity 1C.

In the picture, the L-shape and the rectangle have the same squares. The L-shape has a² − b², the rectangle has (a + b)(a − b).

Notes

(a + b)(a − b) = a² − b²: the product of the sum and the difference is the difference of the squares.

Check yourself

Use the identity to find 87 × 73. Write it as (80 + 7)(80 − 7).

Answer: 6351

(80 + 7)(80 − 7) = 80² − 7² = 6400 − 49 = 6351.

Which is the expansion of (x + 5)(x − 5)?

Spot the pattern: 12 × 12 − 5 × 5 = 17 × □. What number goes in the box?

Answer: 7

(12 + 5)(12 − 5) = 17 × 7 = 119, and 144 − 25 = 119.

Why do the middle terms vanish in (a + b)(a − b)?

• x² − 25 — correct. Yes! x² − 5x + 5x − 25 = x² − 25. The middle terms cancel.
• x² − 10x + 25. That is (x − 5)². In (x + 5)(x − 5) the terms −5x and +5x cancel.
• x² + 25. The last term is 5 × (−5) = −25, not +25.
• Because a and b are equal. a and b can be any numbers. The identity does not need a = b.
• ab and ba are the same number. One is taken away and one is added — correct. Yes! −ab + ba = 0, so only a² − b² is left.
• Because a − b is zero. a − b is zero only when a = b. The identity works for every a and b.