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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
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