Question

Solve the following equation by factorization:

`2(x/(x + 1))^2 - 5(x/(x + 1)) + 2 = 0, x ≠ -1`

Answer 1

Let us consider `y = x/(x + 1)`.

Substituting `y = x/(x + 1)` in equation `2(x/(x + 1))^2 - 5(x/(x + 1)) + 2 = 0`, we get:

⇒ 2y² – 5y + 2 = 0

⇒ 2y² – 4y – y + 2 = 0

⇒ 2y(y – 2) – 1(y – 2) = 0

⇒ (2y – 1)(y – 2) = 0

⇒ (2y – 1) = 0 or (y – 2) = 0   ...[Using zero-product rule]

⇒ 2y = 1 or y = 2

⇒ y = `1/2` or y = 2

Now we have,

Case 1: `y = 1/2`

⇒ `y = x/(x + 1)`

⇒ `1/2 = x/(x + 1)`

⇒ x + 1 = 2x

⇒ 2x – x = 1

⇒ x = 1

Case 2: y = 2

⇒ `y = x/(x + 1)`

⇒ `2 = x/(x + 1)`

⇒ 2(x + 1) = x

⇒ 2x + 2 = x

⇒ 2x – x = –2

⇒ x = –2

Hence, x = {–2, 1}.