Squares of two-term expressions

The idea

To square a binomial expression, take its two terms as a and b in (a + b)² = a² + 2ab + b² and simplify each term.

Whole terms as a and b

The identity is a pattern, and a and b can be anything: numbers, letters, or terms like 5x. How would you square 5x + 2y?

In (5x + 2y)², what is a?

• 5x
• x
• 5

Expand it

The two terms of the bracket are a and b. Work out a², then 2ab, then b².

a and b are terms

To square a binomial, take its two terms as a and b in (a + b)² = a² + 2ab + b², and simplify each term.

(5x + 2y)² = (5x)² + 2(5x)⁠(2y) + (2y)² = 25x² + 20xy + 4y² Put the whole term in brackets before you square it: (5x)² = 25x², not 5x².

Notes

To square a binomial, take its two terms as a and b, then simplify a² + 2ab + b² term by term.

Check yourself

In (3m + 4n)², what are a and b?

(x + 6)² = x² + ?x + 36. What number goes in the box?

Answer: 12

2ab = 2 × x × 6 = 12x.

Expand (2a + 3b)².

In the expansion of (4x + 5y)², what is the coefficient of xy?

Answer: 40

2 × 4x × 5y = 40xy.

• a = 3m and b = 4n — correct. Yes. The two terms of the bracket are a and b.
• a = 3 and b = 4. The letters belong to the terms: a = 3m and b = 4n.
• a = m and b = n. The numbers belong to the terms too.
• 2a² + 6ab + 3b². (2a)² = 4a², not 2a². Square the number as well as the letter.
• 4a² + 12ab + 9b² — correct. Yes: (2a)² = 4a², 2(2a)⁠(3b) = 12ab and (3b)² = 9b².
• 4a² + 9b². The middle term 2(2a)⁠(3b) = 12ab is missing.