Section 1.1 - Rational Equations that Reduce to Linear or Quadratic

Section Objectives

1. Determine the values of the variable that are restricted from a rational expression.
2. Solve special cases of rational equations that reduce to linear or quadratic equations.

We have already discussed the solution of certain types of rational equations. We are going to extend our list of "certain types" by considering equations with quadratic denominators. These equations will typically involve factoring.

Examples

• What value of $x$ is restricted from the expression $\displaystyle \frac{x-6}{x^2-12x+36}$?

• What value of $t$ is restricted from the expression $\displaystyle \frac{t^2+3t-18}{t^2+15t+54}$?

• Solve for $w$: $\quad \displaystyle \frac{1}{w-4}+\frac{3}{w+2} = \frac{10}{w^2-2w-8}$

• Solve for $v$: $\quad \displaystyle v + \frac{2}{v} = 8 -\frac{13}{v}$

• Solve for $x$: $\quad \displaystyle 4 + \frac{3}{x-2} = \frac{3}{(x-1)(x-2)}$

• Solve for $x$: $\quad \displaystyle \frac{x-4}{x-5}-1=\frac{x-2}{x-6}$