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

Solve the following equation.

\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\}