Question

Solve: `2sqrt(x/"a") + 3sqrt("a"/x) = "b"/"a" + (6"a")/"b"`

Answer 1

Put `sqrt(x/"a")` = y

`(2 sqrt(x)/sqrt("a") + 3 sqrt("a")/sqrt(x))^2 = ("b"/"a" + (6"a")/"b")^2`

`4 x/"a" + 9"a"/x + 12 = "b"^2/"a"^2 + (6^2"a"^2)/"b"^2 + 12`

Multiplying this equation by `"a"/"b" x`

`("a"/4 x)(4/"a") x + ("a"/4 x) ((9"a")/x) = ("a"/4 x) "b"^2/"a"^2 + ("a"/4) + (6^2"a"^2)/"b"^2`

`x^2 + (9"a"^2)/4 = ("b"^2x)/(4"a") + (36"a"^33x)/(4"b"^2)`

`x^2 - ("b"^2x)/(4"a") + (9"a"^2)/4 - (9"a"^3x)/"b"^2` = 0

`x(x - "b"^2/(4"a")) - (9"a"^3)/"b"^2 (x - "b"^2/(4"a"))` = 0

`(x - "b"^2/(4"a"))(x - (9"a"^3)/"b"^2)` = 0

x  = `("b"^2)/(4"a")` or x = `(9"a"^2)/"b"^2`

The roots are `"b"^2/(4"a"), (9"a"^3)/"b"^2`