The perfect square identities give you a formula for squaring any two-term expression without expanding every step from scratch. There are two: (a + b)² = a² + 2ab + b² for a sum, and (a − b)² = a² − 2ab + b² for a difference. Recognising which one applies speeds up GCSE expanding, factorising and mental arithmetic.

What are the two perfect square identities?

An algebraic identity is an equation that is true for every possible value of its variables, not just particular ones. The two perfect square identities are:

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab + b²

The middle term is 2ab with a + sign for the addition identity and a − sign for the subtraction identity. This is the part students most commonly get wrong.

How do you prove these identities?

Both identities follow from expanding the brackets:

Proof of (a + b)²:

(a + b)² = (a + b)(a + b)

= a(a + b) + b(a + b)

= a² + ab + ba + b²

= a² + 2ab + b² ✓

Proof of (a − b)²:

(a − b)² = (a − b)(a − b)

= a(a − b) − b(a − b)

= a² − ab − ba + b²

= a² − 2ab + b² ✓

The key step in the second proof: −b × (−b) = +b², because a negative times a negative is positive.

How do you use the identity to expand a perfect square?

Identify a and b from the expression, substitute into the identity, and write the three terms.

Worked examples:

Expression a b Result using a² + 2ab + b²
(x + 5)² x 5 x² + 10x + 25
(x − 3)² x 3 x² − 6x + 9
(2x + 1)² 2x 1 4x² + 4x + 1
(3x − 4)² 3x 4 9x² − 24x + 16

Detail for (3x − 4)²: a = 3x, b = 4.

  • a² = (3x)² = 9x²
  • 2ab = 2 × 3x × 4 = 24x
  • b² = 16
  • Answer: 9x² − 24x + 16 ✓

How do the identities speed up mental arithmetic?

Because a and b can be numbers, the identities turn squaring into three easier steps.

Example 1: Calculate 51².

Write 51 = 50 + 1. Use (a + b)² with a = 50, b = 1:

51² = 50² + 2 × 50 × 1 + 1² = 2500 + 100 + 1 = 2601

Example 2: Calculate 99².

Write 99 = 100 − 1. Use (a − b)² with a = 100, b = 1:

99² = 100² − 2 × 100 × 1 + 1² = 10 000 − 200 + 1 = 9801

Both calculations take seconds once you know which identity to use.

How do you recognise a perfect square trinomial?

A trinomial ax² + bx + c is a perfect square if:

  • The first and last terms are perfect squares themselves, and
  • The middle term equals twice the product of their square roots.

Check whether x² + 6x + 9 is a perfect square:

  • √(x²) = x; √9 = 3.
  • Twice their product: 2 × x × 3 = 6x. ✓
  • So x² + 6x + 9 = (x + 3)².

Check whether x² − 8x + 16 is a perfect square:

  • √(x²) = x; √16 = 4.
  • Twice their product: 2 × x × 4 = 8x. ✓ (middle term is −8x, matching (a − b)²)
  • So x² − 8x + 16 = (x − 4)².

Check whether x² + 5x + 9 is a perfect square:

  • √(x²) = x; √9 = 3.
  • Twice their product: 2 × x × 3 = 6x ≠ 5x. ✗
  • Not a perfect square.

What mistakes should you avoid?

Mistake 1 — Writing (a + b)² = a² + b². This is the single most common algebra error. The middle term 2ab is always present. There is no shortcut that skips it: (a + b)² ≠ a² + b².

Mistake 2 — Getting the sign of the middle term wrong. (a − b)² has a negative middle term: a² − 2ab + b². Both the first and last terms are positive (squaring removes the minus), but the middle term is negative.

Mistake 3 — Forgetting to square the coefficient when a = kx. In (3x)², the coefficient squares too: (3x)² = 9x², not 3x².

Frequently asked questions

Is (a + b)² = a² + b² ever true?

Only when ab = 0 — i.e., when a = 0 or b = 0, making the middle term 2ab = 0. For any two non-zero values, (a + b)² ≠ a² + b². For example, (3 + 4)² = 49 but 3² + 4² = 25. The difference (24) is exactly 2 × 3 × 4 = 2ab.

How does (a + b)² relate to the area of a square?

Draw a square with side a + b. Divide it into four rectangles by drawing a line a from the left and a line a from the top: one square of side a (area a²), one square of side b (area b²), and two rectangles of dimensions a × b (area ab each). Total area = a² + 2ab + b² — exactly the identity. This gives a geometric proof.

When would I use these identities at GCSE instead of just expanding?

Whenever a question contains an obvious perfect square such as (x + 5)² or (2x − 3)², writing the three-term result directly from the identity is faster and less error-prone than expanding step by step. They also appear in completing the square, recognising factors in quadratic expressions and simplifying surds.

What is (a + b)(a − b)?

This is a different identity: (a + b)(a − b) = a² − b², called the difference of two squares. The middle terms +ab and −ab cancel. It is useful for factorising expressions like x² − 25 = (x + 5)(x − 5), and worth memorising alongside the two perfect square identities.

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