Question

Solve the following equation by factorisation method:

`((4x - 3)/(2x + 1)) - 10((2x + 1)/(4x - 3)) = 3, x ≠ -1/2, 3/4`

Solve the following quadratic equation:

`((4x - 3)/(2x + 1)) - 10((2x + 1)/(4x - 3)) = 3, x ≠ (-1)/2, 3/4` 

Answer 1

Given:

`((4x - 3)/(2x + 1)) - 10((2x + 1)/(4x - 3)) = 3`

Putting `(4x - 3)/(2x + 1) = y`, we get: 

`y - 10/y = 3` 

⇒ `(y^2 - 10)/y = 3` 

⇒ y² – 10 = 3y   ...[On cross multiplying] 

⇒ y² – 3y – 10 = 0 

⇒ y² – (5 – 2)y – 10 = 0 

⇒ y² – 5y + 2y – 10 = 0 

⇒ y(y – 5) + 2(y – 5) = 0 

⇒ (y – 5)(y + 2) = 0 

⇒ y – 5 = 0 or y + 2 = 0 

⇒ y = 5 or y = –2 

Case I: 

If y = 5, we get: 

`(4x - 3)/(2x + 1) = 5`  

⇒ 4x – 3 = 5(2x + 1)   ...[On cross multiplying] 

⇒ 4x – 3 = 10x + 5 

⇒ –6x = 8 

⇒ `-6x=8` 

⇒ `x = 8/6` 

⇒ `x = -4/3` 

Case II: 

If y = –2, we get: 

`(4x - 3)/(2x + 1) = 2` 

⇒ 4x – 3 = –2(2x + 1) 

⇒ 4x – 3 = –4x – 2 

⇒ 8x = 1 

⇒ `x = 1/8` 

Hence, the roots of the equation are `-4/3` and `1/8`.