Question

Solve the following equation.

`(x^2 + 12x - 20)/(3x - 5) = (x^2 + 8x + 12)/(2x + 3)`

Answer 1

`(x^2 + 12x -20)/( 3x - 5 ) = (x^2 + 8x + 12)/( 2x + 3 )`

Multiplying both sides by `1/4`, we get

`(x^2 + 12x -20)/( 12x - 20 ) = (x^2 + 8x + 12)/( 8x + 12 )`

Using dividendo, we get

`[(x^2 + 12x -20) - ( 12x - 20 )]/( 12x - 20 ) = [(x^2 + 8x + 12) - ( 8x + 12 )]/ ( 8x + 12 )`

⇒ `[x^2 + 12x -20 - 12x + 20 ]/( 12x - 20 ) = [x^2 + 8x + 12 -  8x - 12 ]/ ( 8x + 12 )`

⇒ `x^2/( 12x - 20 ) = x^2/ ( 8x + 12 )`

This equation is true for x = 0.

Therefore, x = 0 is a solution of the given equation.

If x ≠ 0, then x² ≠ 0.

Dividing both sides by x², we get

`1/(12x - 20)= 1/(8x + 12)`

⇒ 12x - 20 = 8x + 12

⇒ 12x - 8x = 20 + 12 

⇒ 4x = 32

⇒ x = 8

Thus, the solutions of the given equation are x = 0 and x = 8.