Binomial Squares

Statement

The binomial square formulas expand the square of a sum or difference:

\[(a+b)^2 = a^2 + 2ab + b^2 \quad\text{and}\quad (a-b)^2 = a^2 - 2ab + b^2\]

These results are used in algebra to simplify expressions, solve equations, and factorise quadratics.

Why it’s true (short reason)

• Squaring means multiplying by itself.
• \((a+b)^2 = (a+b)(a+b)\).
• Expand using distributive law: \(a^2 + ab + ab + b^2 = a^2 + 2ab + b^2\).
• Similarly, \((a-b)^2 = (a-b)(a-b) = a^2 - ab - ab + b^2 = a^2 - 2ab + b^2\).

Recipe (how to use it)

1. Identify whether you have \((a+b)^2\) or \((a-b)^2\).
2. Square the first term: \(a^2\).
3. Multiply the two terms and double it: \(±2ab\).
4. Square the last term: \(b^2\).
5. Write down the result in simplified form.

Spotting it

You use this formula when you see:

• A squared bracket, like \((x+3)^2\) or \((2y-5)^2\).
• A quadratic in the form \(a^2 ± 2ab + b^2\), which can be factorised back into a square.

Common pairings

• Completing the square (turning quadratics into squared brackets).
• Factorising perfect square trinomials.
• Expanding brackets in algebra and simplification tasks.

Mini examples

1. \((x+4)^2 = x^2 + 8x + 16\).
2. \((y-7)^2 = y^2 - 14y + 49\).

Pitfalls

• Forgetting the middle term (thinking \((a+b)^2 = a^2+b^2\)).
• Getting the sign wrong in \((a-b)^2\).
• Not squaring the second term correctly (e.g. \((3)^2=9\), not 6).

Exam strategy

• Write out \((a±b)(a±b)\) if unsure.
• Check each expansion carefully—three terms should always appear.
• Remember symmetry: the first and last terms are squares, the middle term is double the product.

Summary

The binomial squares are key identities in algebra. They simplify expansions, support factorisation, and are a stepping stone towards completing the square and quadratic formula applications.