Q4 If (x-1/2)² – (x-3/2)² = x+2 , then the value of x:

• (A) 2
• (B) 4
• (C) 6
• (D) 8

Solve (x-1)² – (x-2)² = 2: Complete Step-by-Step Solution

The equation (x−1)² − (x−2)² = 2 simplifies to x = 2.5 using the difference of squares identity, but none of the given options (2, 4, 6, 8) satisfy it exactly. Among them, x=4 yields the closest value of 5.

Solution Steps

1. Apply the difference of squares formula: a² − b² = (a + b)(a − b) where a = x − 1 and b = x − 2.
2. This gives: (x − 1 + x − 2)(x − 1 − (x − 2)) = 2
3. Simplifying: (2x − 3)(1) = 2
4. Thus: 2x − 3 = 2, so 2x = 5 and x = 5/2 = 2.5
5. Verification: (2.5 − 1)² − (2.5 − 2)² = 2.25 − 0.25 = 2 ✅

Option Analysis

Option	x Value	Calculation	Result
(A)	2	(2−1)²−(2−2)²=1−0=1	≠2
(B)	4	(4−1)²−(4−2)²=9−4=5	≠2 (closest)
(C)	6	(6−1)²−(6−2)²=25−16=9	≠2
(D)	8	(8−1)²−(8−2)²=49−36=13	≠2

Introduction to Solving (x-1)^2 – (x-2)^2 = 2

Mastering equations like solve (x-1)^2 – (x-2)^2 = 2 is essential for algebra students preparing for competitive exams. This problem leverages the difference of squares identity a²−b²=(a+b)(a−b) to quickly find the value of x without expansion.

Detailed Solution Using Algebraic Identity

Set a=x−1 and b=x−2. The equation becomes (a)²−(b)²=2, or (a+b)(a−b)=2. Substitute: a+b=(x−1)+(x−2)=2x−3 and a−b=(x−1)−(x−2)=1. Thus, (2x−3)(1)=2, so 2x−3=2, 2x=5, and x=2.5.

Why Difference of Squares Works Best

Expanding fully: (x²−2x+1)−(x²−4x+4)=2x−4=2, so 2x=6, x=3. The identity method is superior, error-free, and faster for competitive exams like CSIR NET.