Question

Solve the following equation by factorization:

4x² – 4ax + (a² – b²) = 0, where a, b ∈ R

[Hint: Given equation may be written as: 4x² – 2(a + b)x – 2(a – b)x + (a² – b²) = 0]

Answer 1

Given,

⇒ 4x² – 4ax + (a² – b²) = 0

⇒ (4x² – 4ax + a²) – b² = 0

⇒ [(2x)² – 2 × a × 2x + (a)²] – b² = 0

⇒ (2x – a)² – b² = 0

⇒ (2x – a + b)(2x – a – b) = 0

⇒ (2x – a + b) = 0 or (2x – a – b) = 0   ...[Using zero-product rule]

⇒ 2x = a – b or 2x = a + b

⇒ `x = (a - b)/2` or `x = (a + b)/2`

Hence, `x = {(a + b)/2, (a - b)/2}`.