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Chapter 1.6

Chapter 1.6. Other Types of Equations. Rational Equations

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Chapter 1.6

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  1. 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 # 1 - 78

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