Question

Question #9042c

Answer 1

Solve the equation for y = 3.

Explanation:

Firstly, simplify `x^(-1/2) - 5x^(-1/4) + 9 = 3`
Simplifying, we get `1/(sqrt(x)) - 5/(x^(1/4)) = 3 - 9 = -6`
Simplify both fractions containing x,
`1/(sqrt(x)) - (5*(x^(1/4)))/(x^(1/4) * (x^(1/4))) = -6`
`1/(sqrt(x)) - (5x^(1/4))/sqrt(x) = -6`
`(1 - 5x^(1/4))/sqrt(x) = -6`
`(1 - 5x^(1/4)) = -6sqrt(x)`
`6sqrt(x) - 5x^(1/4) + 1 = 0`
let `a^2 = sqrt(x), a = x^(1/4)`
`6a^2 - 5a + 1 = 0`
Factorise:
`6a^2 - 5a + 1 = 0`
`(3a-1)(2a-1) = 0`
Thus `a = 1/3 or a = 1/2`
Since `a = x^(1/4), a^2 = sqrt(x) = 1/9, a^4 = x, x = 1/81`
and since `a^4 = x, x = 1/2^4`
So `x = 1/16 or x = 1/81`
graph{x^(-1/2) - 5x^(-1/4) + 9 [-0.2597, 0.3647, 2.7107, 3.023]}