Question

Factorise the following:

x² – 25y² + 2x – 10y

Answer 1

Given expression: x² – 25y² + 2x – 10y

Step-wise calculation:

1. Group terms: (x² + 2x) – (25y² + 10y)

2. Complete the square for each group.

For (x² + 2x):

x² + 2x = (x² + 2x + 1) – 1

x² + 2x = (x + 1)² – 1

For (25y² + 10y):

`25y^2 + 10y = 25(y^2 + 2/5 y)`

`25y^2 + 10y = 25(y^2 + 2/5 y + (1/5)^2 - (1/5)^2)`

`25y^2 + 10y = 25((y + 1/5)^2 - 1/25)`

`25y^2 + 10y = 25(y + 1/5)^2 - 1`

3. Substitute these back:

`(x + 1)^2 - 1 - [25(y + 1/5)^2 - 1]`

= `(x + 1)^2 - 1 - 25(y + 1/5)^2 + 1`

= `(x + 1)^2 - 25(y + 1/5)^2`

4. Recognize this as a difference of squares:

a² – b² = (a + b)(a – b) where a = (x + 1), `b = 5(y + 1/5) = 5y + 1`

5. So factorization is:

(x + 1 + 5y + 1)(x + 1 – 5y – 1) = (x + 5y + 2)(x – 5y)