Question

Solve the following equation using quadratic formula:

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

Answer 1

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

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

⇒ `(x^2 - 4x - x + 4 + (x^2 - 2x - 3x + 6))/(x^2 - 4x - 2x + 8) = 10/3`

⇒ `(x^2 - 5x + 4 + (x^2 - 5x + 6))/(x^2 - 6x + 8) = 10/3`

⇒ `(2x^2 - 10x + 10)/(x^2 - 6x + 8) = 10/3`

⇒ 3(2x² – 10x + 10) = 10(x² – 6x + 8)

⇒ 6x² – 30x + 30 = 10x² – 60x + 80

⇒ 10x² – 60x + 80 – (6x² – 30x + 30) = 0

⇒ 10x² – 60x + 80 – 6x² + 30x – 30 = 0

⇒ 4x² – 30x + 50 = 0

⇒ 2(2x² – 15x + 25) = 0

⇒ 2x² – 15x + 25 = 0

Comparing equation 2x² – 15x + 25 = 0 with ax² + bx + c = 0, we get:

a = 2, b = –15 and c = 25

By formula,

`x = (-b ± sqrt(b^2 - 4ac))/(2a)`

Substituting values we get:

⇒ `x = (-(-15) ± sqrt((-15)^2 - 4(2)(25)))/(2(2))`

= `(15 ± sqrt(225 - 200))/4`

= `(15 ± sqrt(25))/4`

= `(15 ± 5)/4`

= `(15 + 5)/4` or `(15 - 5)/4`

= `20/4` or `10/4`

= 5 or `5/2`

Hence, `x = {5, 5/2}`.