Structure Recognition Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Recognise and name the algebraic structure in x2−6x+9x^2 - 6x + 9, then factorise.

Solution

  1. 1
    Compare with the perfect square pattern: (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2.
  2. 2
    Here: a2=x2a^2 = x^2 gives a=xa = x; b2=9b^2 = 9 gives b=3b = 3; and 2ab=6x2ab = 6x — consistent.
  3. 3
    Therefore x2−6x+9=(x−3)2x^2 - 6x + 9 = (x-3)^2.

Answer

(x−3)2(x-3)^2
Structure recognition means identifying a known pattern (here, a perfect square trinomial) within an expression. Once recognised, the factorisation follows immediately.

About Structure Recognition

The skill of identifying that a given mathematical expression or problem belongs to a known family or matches a recognizable pattern.

Learn more about Structure Recognition →

More Structure Recognition Examples

Example 2 medium

Identify the structure in [formula] and use it to simplify [formula].

Example 3 easy

Identify the structure and factorise: [formula].

Example 4 medium

Recognise the structure in the sum [formula] and compute it using the geometric series formula.

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