Presentation on theme: "Other Types of Equations"— Presentation transcript:
1 Other Types of Equations Chapter 1.6Other Types of Equations
2 Rational EquationsA 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 = 0is not a quadratic equation because of the x4 term. However, with substitutionsu = x2 and u2 = x4the equation becomes12u2 – 11u + 2 = 0which 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