# Equations Involving One Variable Leading to Quadratic Equation| Algebra| ENGG10 LE3| Problem 4

#### $\frac{1}{x}-\frac{11}{\sqrt{x}}+18=0$

SOLUTION:

Let $u=\frac{1}{\sqrt{x}}$; so $u^2=\frac{1}{x}$.

Substitute into the original equation.

$u^2-11u+18=0$

Solve for u using factoring

$\left(u-9\right)\left(u-2\right)=0$

$u=9$ or $u=2$

We have 2 values of u. When $u=9$, we have

$\frac{1}{\sqrt{x}}=9$

$\sqrt{x}=\frac{1}{9}$

$x=\frac{1}{81}$

When $u=2$, we have

$\frac{1}{\sqrt{x}}=2$

$\sqrt{x}=\frac{1}{2}$

$x=\frac{1}{4}$

Since both $\frac{1}{81}\:and\:\frac{1}{4}$ are in the domain of the equation, the solution set is

$\left\{\frac{1}{81},\:\frac{1}{4}\right\}$