Solve: [tex]\frac{1}{x} + 6 = \frac{5}{\sqrt{x}}[/tex]
Start by squaring both sides to eliminate the square root.
[tex](\frac{1}{x} + 6)^{2} = \frac{25}{x}[/tex]
Expand the left hand side.
[tex]\frac{1}{x^{2}} + \frac{12}{x} + 36 = \frac{25}{x}[/tex]
Simplify.
[tex]\frac{1}{x^{2}} - \frac{13}{x} + 36 = 0[/tex]
[tex]\frac{1 - 13x + 36x^{2}}{x^{2}} = 0[/tex]
Solve for x.
[tex]36x^{2} - 13x + 1 = 0[/tex]
[tex]36x^{2} - 9x - 4x + 1 = 0[/tex]
[tex]9x(4x - 1) - (4x - 1) = 0[/tex]
[tex](9x - 1)(4x - 1) = 0[/tex]
[tex]\therefore x = \frac{1}{9}, \frac{1}{4}[/tex]
