Question

Solve the following by reducing them to quadratic equations:
`sqrt(x/(1 -x)) + sqrt((1 - x)/x) = (13)/(6)`.

Answer 1

Given equation `sqrt(x/(1 -x)) + sqrt((1 - x)/x) = (13)/(6)`
Putting `sqrt(x/(1 - x)) = y,` then given equation reducible to the form `y + (1)/y = (13)/(6)`
⇒ `(y^2 + 1)/y = (13)/(6)`
⇒ 6y² + 6 = 13y
⇒ 6y² - 13y + 6 = 0
⇒ 6y² - 9y - 4y + 6 = 0
⇒ 3y(2y - 3) -2(2y - 3) = 0
⇒ (2y - 3) (3y - 2) = 0
⇒ 2y - 3 = 0 or 3y - 2 = 0
⇒ y = 3/2 or y = 2/3
But `sqrt(x/(1 - x)) = y`
∴ `sqrt(x/(1 - x)) = (3)/(2)`
Squaring `x/(1 -x) = (9)/(4)`
⇒ 4x = 9 - 9x
⇒ 13x = 9
⇒ x = `(9)/(13)`
or
`sqrt(x/(1 - x)) = (2)/(3)`
Squaring `x/(1 - x) = (4)/(9)`
⇒ 9x = 4 - 4x
⇒ 9x + 4x = 4
⇒ 13x = 4
⇒ x = `(4)/(13)`
Hence, the required roots are `{9/13,4/13}`.