Question

Factorise the following:

(ax – by)² – (bx – ay)²

Answer 1

Given expression: (ax – by)² – (bx – ay)²

Step-wise calculation

Step 1: Recognize this as a difference of squares:

A² – B² = (A + B)(A – B), where A = (ax – by), B = (bx – ay)

Step 2: Apply the formula:

(ax – by)² – (bx – ay)²

= ((ax – by) + (bx – ay)) × ((ax – by) – (bx – ay))

Step 3: Simplify each factor:

(ax – by) + (bx – ay) 

= ax – by + bx – ay

= (ax – ay) + (bx – by)

= a(x – y) + b(x – y)

= (a + b)(x – y)

(ax – by) – (bx – ay) 

= ax – by – bx + ay 

= (ax – bx) + (ay – by)

= x(a – b) + y(a – b)

= (a – b)(x + y)

Step 4: Therefore,

(ax – by)² – (bx – ay)²

= (a + b)(x – y) × (a – b)(x + y)

Step 5: Rearranging the factors:

(a + b)(a – b)(x – y)(x + y)