Question #9042c

1 Answer
Apr 14, 2017

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]}