Squaring with (a + b)²

In (a + b)², a and b can be whole terms, not just single letters. Think of them as two boxes: whatever goes in the first box is 'a', whatever goes in the second is 'b'.

Example 3. (5x + 2y)² with a = 5x and b = 2y:

• first squared: (5x)² = 25x²;
• twice the product: 2(5x)(2y) = 20xy;
• second squared: (2y)² = 4y².

So (5x + 2y)² = 25x² + 20xy + 4y².

Watch the middle term. It is twice the product of the two terms. Forgetting the 2 is the most common slip. And square the whole term: (5x)² is 25x², not 5x².

Fractions work the same way: (x + 1/(2y))² = x² + 2 × x × 1/(2y) + 1/(4y²) = x² + x/y + 1/(4y²).

Mental maths. Split a number into an easy round part and a small part. Example 4: 43² = (40 + 3)² = 1600 + 240 + 9 = 1849.