The algebraic identity (a – b)² is one of the most commonly used formulas in mathematics. It helps students simplify expressions quickly without doing lengthy multiplication each time. This identity plays a major role in algebra, factorisation, quadratic equations, mental calculations, and exam-level problem-solving. Because it saves time and reduces mistakes, competitive exams like CUET, SSC, JEE, and Banking frequently test this identity. Understanding how (a – b)² expands makes solving expressions easier and faster.
Formula for (a – b)² - Overview
| Formula | Variables | When It Is Used |
|---|---|---|
| (a – b)² = a² – 2ab + b² | a = first term, b = second term | Used in algebraic expansions, simplification, quadratic expressions |
What is (a – b)² in Maths?
In mathematics, (a – b)² represents the square of a binomial where the two terms are separated by subtraction. It is the expanded form of multiplying a binomial with itself. Instead of solving (a–b)(a–b)(a–b)(a–b)(a–b)(a–b) manually, we use the direct identity:
(a – b)² = a² – 2ab + b²
This identity comes from expanding each term and combining like terms. When you multiply:
(a–b)(a–b)(a – b)(a – b)(a–b)(a–b)
You get:
- a×a=a2a × a = a²a×a=a2
- a×(–b)=–aba × (–b) = –aba×(–b)=–ab
- (–b)×a=–ab(–b) × a = –ab(–b)×a=–ab
- (–b)×(–b)=b2(–b) × (–b) = b²(–b)×(–b)=b2
Combining them:
a2–ab–ab+b2=a2–2ab+b2a² – ab – ab + b² = a² – 2ab + b²a2–ab–ab+b2=a2–2ab+b2
This identity is used in simplification, geometry problems, mental maths shortcuts, and exam questions where quick expansion is needed.
Examples to Calculate (a – b)²
Example 1: Expand (6 – 4)²
Step 1: Use the identity → (a – b)² = a² – 2ab + b²
Step 2: a = 6, b = 4
Step 3: = 6² – 2(6)(4) + 4²
Step 4: = 36 – 48 + 16
Result: 4
Therefore, (6 – 4)² = 4.
Example 2: Expand (x – 9)²
Step 1: Apply the identity → (a – b)² = a² – 2ab + b²
Step 2: a = x, b = 9
Step 3: = x² – 2(x)(9) + 9²
Step 4: = x² – 18x + 81
Result: x² – 18x + 81
Therefore, (x – 9)² = x² – 18x + 81.
FAQs About (a – b)² Formula
Q1. Why is the middle term negative in (a – b)²?
Because multiplying a with –b gives –ab twice, so the combined middle term becomes –2ab.
Q2. Is (a – b)² the same as a² – b²?
No. (a – b)² = a² – 2ab + b², while a² – b² is a different identity.
Q3. Where is the (a – b)² formula used in real life?
In algebraic simplification, geometry, mental maths, and pattern-based reasoning.
Q4. Is this identity important for competitive exams?
Yes. It appears frequently in CUET, JEE, SSC, Banking, and school-level question papers.
Q5. Can we apply (a – b)² to fractions or negative numbers?
Yes. The identity works for all real numbers, including decimals, fractions, and negatives.