1. Other Types of Equations
Chapter 1.6
Other Types of Equations

2. Rational Equations
A rational equation is an equation that has a rational expression for one or more terms. Since a rational expression is not defined when its denominator is 0, values of the variable for which any denominator equals 0 cannot be solutions of the equations. To solve a rational equation, begin by multiplying both sides by the least common denominator (LCD) of the terms of the equation.

3. Example 1 Solving Rational Equations That Lead to Linear Equations
Solve each equation.

4. Example 1 Solving Rational Equations That Lead to Linear Equations
Solve each equation.

5. Example 1 Solving Rational Equations That Lead to Quadratic Equations
Solve each equation.

6. Example 1 Solving Rational Equations That Lead to Quadratic Equations
Solve each equation.

7. To solve an equation such as
in which the variable appears in a radicand, we use the following power property to eliminate the radical.

8. If P and Q are algebraic expressions, then every solution of the equation P = Q is also a solution of the equation Pn = Qn, for any positive integer n.

9. We also use the power property to solve equations such as
where the variable appears in an expression that is the base of a term with a rational exponent.

10. Example 3 Solving an Equation Containing a Radical (Square Root)
Solve

11. Example 4 Solving an Equation Containing Two Radicals
Solve

12. Example 5 Solving an Equation Containing A Radical (Cube Root)
Solve

13. Equations Quadratic in Form
Many equations that are not quadratic equations can be solved by the methods discussed in Section 1.4.

14. The equation 12x4 – 11x2 + 2 = 0
is not a quadratic equation because of the x4 term. However, with substitutions
u = x2 and u2 = x4
the equation becomes
12u2 – 11u + 2 = 0
which is a quadratic equation in u. This quadratic equation can be solved to find u, then u = x2 can be used to find the values of x

15. Example 6 Solving an Equation Quadratic in Form
Solve 12x4 – 11x2 + 2 = 0

16. Example 7 Solving an Equation Quadratic in Form
Solve each equation

17. Example 8 Solving an Equation That Leads to One That is Quadratic in Form
Solve each equation

18. Section 1.6 #