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**Other Types of Equations**

Chapter 1.6 Other Types of Equations

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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.

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**Example 1 Solving Rational Equations That Lead to Linear Equations**

Solve each equation.

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**Example 1 Solving Rational Equations That Lead to Linear Equations**

Solve each equation.

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**Example 1 Solving Rational Equations That Lead to Quadratic Equations**

Solve each equation.

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**Example 1 Solving Rational Equations That Lead to Quadratic Equations**

Solve each equation.

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**To solve an equation such as**

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

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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.

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**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.

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

Solve

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**Example 4 Solving an Equation Containing Two Radicals**

Solve

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

Solve

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**Equations Quadratic in Form**

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

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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

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**Example 6 Solving an Equation Quadratic in Form**

Solve 12x4 – 11x2 + 2 = 0

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**Example 7 Solving an Equation Quadratic in Form**

Solve each equation

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

Solve each equation

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Section 1.6 #

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