The expression (a–b)2(a – b)²(a–b)2 is a fundamental algebraic identity used to simplify binomials quickly. Instead of multiplying (a–b)(a–b)(a – b)(a – b)(a–b)(a–b) manually, this identity gives a direct and efficient way to expand expressions. It is widely used in algebra, factorization, quadratic equations, mental maths, and competitive exam questions. Understanding this formula helps students save time, reduce errors, and solve problems more confidently.
Formula for Calculating (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, simplifications, and exam-based problems |
What is (a – b)² in Maths?
In mathematics, (a–b)2(a – b)²(a–b)2 is an algebraic identity that expands the square of a binomial containing subtraction. Instead of performing the longer multiplication process, we use the identity directly to get the expanded form. It simplifies calculations, especially in algebra, factorization, and solving quadratic expressions.
To understand how it expands, consider:
(a–b)(a–b)(a – b)(a – b)(a–b)(a–b)
Now multiply term by term:
- 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 these:
a2−ab−ab+b2=a2−2ab+b2a² - ab - ab + b² = a² - 2ab + b²a2−ab−ab+b2=a2−2ab+b2
This identity is essential in solving equations, area problems in geometry, and competitive exams like CUET, SSC, Banking, and JEE, where quick simplification is required.
Examples to Calculate (a – b)²
Example 1: Expand (6 – 4)²
Step 1: Apply the identity → (a – b)² = a² – 2ab + b²
Step 2: Here, a = 6 and 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: Use the formula → (a – b)² = a² – 2ab + b²
Step 2: Here, a = x and 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 while expanding, we get –ab – ab = –2ab, which makes the middle term negative.
Q2. Is (a – b)² the same as a² – b²?
No. Many students confuse the two.
(a – b)² = a² – 2ab + b², not a² – b².
Q3. Where is the (a – b)² formula used in real life?
It is used in algebraic simplifications, geometry, quick calculations, and pattern recognition.
Q4. Is this identity important for competitive exams?
Yes. It is frequently asked in CUET, SSC, Banking, JEE, and many school-level tests.
Q5. Can we apply (a – b)² for negative or fractional values?
Yes. The identity works for all numbers, including negatives, fractions, and decimals.