Chapter 1.6

Other Types of Equations

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.

Solve each equation.

Example 1 Solving Rational Equations That Lead to Linear Equations

xx

xx

1

2

3

13

Solve each equation.

Example 1 Solving Rational Equations That Lead to Linear Equations

22

2

2

xx

x

Solve each equation.

Example 1 Solving Rational Equations That Lead to Quadratic Equations

xxxx

x

2

21

2

232

Solve each equation.

Example 1 Solving Rational Equations That Lead to Quadratic Equations

1

8

1

4

1

42

xxx

x

To solve an equation such as

in which the variable appears in a radicand, we use the following power property to eliminate the radical.

0215 xx

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.

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.

31

31

1234 xx

Solve

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

0215 xx

Solve

Example 4 Solving an Equation Containing Two Radicals

1132 xx

Solve

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

0144 33 2 xxx

Equations Quadratic in Form

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

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

Solve 12x4 – 11x2 + 2 = 0

Example 6 Solving an Equation Quadratic in Form

Solve each equation

Example 7 Solving an Equation Quadratic in Form

0211 31

32

xx

Solve each equation

Example 8 Solving an Equation That Leads to One That is Quadratic in Form

44

2 41

65 xx

Section 1.6 # 1 - 78