Suppose we have to calculate 29². We can express this as (30 − 1)².

Step-by-step solution

Idea: 29 is just below the round number 30, so write it as a difference and use (a − b)² = a² − 2ab + b².

1. 29 = 30 − 1, so a = 30, b = 1.½ mark
2. 29² = (30 − 1)² = 30² − 2 × 30 × 1 + 1²½ mark
3. = 900 − 60 + 1½ mark
4. = 841½ mark

Check: 29 × 29 = 29 × 30 − 29 = 870 − 29 = 841 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Writing (30 − 1)² = 900 − 1 = 899. The middle term −2 × 30 × 1 = −60 is missing.
• Writing the last term as −1. It is +1² = +1.
• Using 29 = 20 + 9 with (a + b)². Correct, but the arithmetic (400 + 360 + 81) is longer than with 30 − 1.

How this can come in the exam

Try one yourself

Find 59² using (a − b)² = a² − 2ab + b².

More questions like this

• Factor completely:
• Find the values of the following using the identity (a − b)² = a² − 2ab + b².
• What will happen if we want to find the square of the sum of three numbers a, b and c, that is, (a + b + c)²?
• Label the squares and rectangles in the figure so that it represents the identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.
• Let us use this identity to find the square of a number, say 119:

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