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Lesson 3 of 8

Exploring Algebraic Identities · Lesson 3 of 8

Factorisation of Algebraic Expressions Using Identities

“Factorisation packs an expression neatly back into its original boxes.”

Learning Objectives

• Recognise perfect-square trinomials. • Factor expressions using square identities. • Take common factors before applying an identity. • Derive and use (a−b)². • Prove the consecutive-square pattern algebraically.

Expansion and factorisation are opposite processes. In expansion, we start with an expression written as a product and multiply its parts to obtain a longer algebraic expression. In factorisation, we do the reverse: we start with the expanded expression and rewrite it as a product of simpler factors.

For example, the identity (a + b)² = a² + 2ab + b² tells us that expanding (a + b)² gives a² + 2ab + b². We can also read the same identity backwards. Whenever an expression has the form a² + 2ab + b², we can factorise it as (a + b)². Such an expression is called a perfect-square trinomial because it can be written as the square of a binomial.

Factor x²+4x+4

Problem
Recognise the identity.

  1. 1.x² is x squared.
  2. 2.4 is 2².
  3. 3.4x=2(x)(2).
  4. 4.Therefore x²+4x+4=(x+2)².
Factor 36x²+12x+1

Problem
Factor completely.

  1. 1.36x²=(6x)².
  2. 2.1=1².
  3. 3.12x=2(6x)(1).
  4. 4.Therefore 36x²+12x+1=(6x+1)².

Common Factor First

Factor 50p²+60pq+18q²

Problem
Factor completely.

  1. 1.Take common factor 2.
  2. 2.=2(25p²+30pq+9q²).
  3. 3.Inside, 25p²=(5p)² and 9q²=(3q)².
  4. 4.30pq=2(5p)(3q).
  5. 5.So the bracket is (5p+3q)².
  6. 6.Final answer: 2(5p+3q)².

Deriving (a−b)²

Replace b with −b in the square-of-a-sum identity.

Square of a differenceLaTeX

Geometric Meaning

Start with a square of side a. Removing strips related to b reduces the side to a−b. The overlap is counted twice during removal and must be added back once, producing the +b² term.

Geometric model of (a minus b) squared A square of side a divided into a smaller square of side a minus b, a removed vertical strip of area ab, and a removed horizontal strip of area b times a minus b. (a − b)² remaining area ab removed b(a − b) removed a − b b a − b b a (a − b)² = a² − ab − b(a − b)
Geometric model of (a-b)^2
Calculate 29²

Problem
Use (a−b)².

  1. 1.29=30−1.
  2. 2.29²=30²−2(30)(1)+1².
  3. 3.=900−60+1=841.

Proving the Consecutive-Square Pattern

General expressionLaTeX
ExpandLaTeX
SimplifyLaTeX

Because n can represent any middle integer, this proves the result for every set of three consecutive squares.

Practice Problems

Practice Problems
  1. Factor 9x²+24xy+16y².
  2. Factor 4s²+20st+25t².
  3. Factor 49x²+28xy+4y².
  4. Evaluate 79² using (a−b)².
  5. Evaluate 193² using (a−b)².
  6. Evaluate 299² using (a−b)².
  7. Explain the proof of the consecutive-square pattern in your own steps.

Key Takeaways

Key Takeaways

• Perfect-square trinomials match a²±2ab+b². • Common factors should be removed first. • (a−b)²=a²−2ab+b². • Identities work both forward and backward. • The consecutive-square pattern can be proved algebraically.

Coming Next

Next, we extend square identities to three terms and study difference of squares.

Previous · Lesson 2

Visualising Identities

Next · Lesson 4

More Identities