Perfect Square Trinomials – Recognising (x + d)² Expanded
A perfect square trinomial is a trinomial x² + bx + c that comes from squaring a binomial: (x + d)² = x² + 2dx + d². You've already seen this expansion direction on the algebraic multiplication pages – this page focuses on going the other way: recognising when a trinomial is secretly a perfect square, and factoring it straight back into (x + d)².
The test is simple: a trinomial x² + bx + c is a perfect square exactly when c equals (b ÷ 2) squared. This single check saves you from trying the general factor-pair method from the previous pages when a faster shortcut is available.
Recognising and Factoring a Perfect Square Trinomial
x² + bx + c is a perfect square trinomial when c = (b ÷ 2)². It factors as (x + d)², where d = b ÷ 2.
(b ÷ 2)² = (10 ÷ 2)² = 5² = 25, which matches c: yes.
d = 12 ÷ 2 = 6. Since 6² = 36 matches c, this factors as (x + 6)².
Real-Life Application
- Completing the square: recognising perfect squares is the core skill behind this key quadratic technique, covered next.
- Physics: perfect square patterns appear in kinetic energy and motion formulas.
- Architecture: some structural formulas simplify neatly when they hide a perfect square.
Key Takeaways
- A perfect square trinomial x² + bx + c has c = (b ÷ 2)².
- It factors directly as (x + d)², where d = b ÷ 2.
- Recognising this pattern is a faster shortcut than general factor-pair factoring.
Practice: Perfect Square Trinomials
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