Other Types of Equations

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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 equation. To solve a rational
  equation, begin by multiplying both sides by the
  least common denominator of the terms of the
  equation.

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Example

• Solve
• Solution The least common denominator is 4(x
  3),which is equal to 0 if x ?3. Therefore, ?3
  cannot possibly be a solution of this equation.
• Check the result we find that the solution set
  is 21.

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Another Example

• Solve
• Solution

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Another Example continued

• Neither proposed solution is valid, so the
  solution set is 0.

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Equations with Radicals

• To solve an equation in which the variable
  appears in the radicand, we use the following
  power property to eliminate the radical.
• Power Property
• 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.
• When using the power property to solve equations,
  the new equation may have more solutions than the
  original equation.
• When an equation contains radicals (or rational
  exponents), it is essential to check all proposed
  solutions in the original equations.

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Radicals

• Solving an Equation Involving Radicals
• Isolate the radical on one side of the equation.
• Raise each side of the equation to a power that
  is the same as the index of the radical so that
  the radical is eliminated.
• If the equation still contains a radical, repeat
  Steps 1 and 2.
• Solve the resulting equation.
• Check each proposed solution in the original
  equation.

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Example

• Solve
• Isolate the radical.
• Square
  both sides of the eqn.
• 
  
• 
  Simplify. The result is a quadratic.
• Put the
  quadratic in standard form.
• Factor.
• 
  Solve for x.

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Example continued

• Check
• If x 9, then
• ?
• 
  
  ?
• As the check shows, only 9 is a solution, giving
  the solution set 9

• Check
• If x 2, then
• ?
• ?

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Solving an Equation Containing Two Radicals

• Example Solve
• Solution

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Solving an Equation Containing Two Radicals
continued

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Solving an Equation Containing Two Radicals
continued

• Check x 7
• ?
• ?

• Check x ?1
• ?
• ?
• The solution set is ?1

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Quadratics

• Equations Quadratic in Form
• An equation is said to be quadratic in form
  if it can be written as
• au2 bu c 0,
• where a ? 0, and u is some algebraic expression.

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Example

• x4 ? 2x2 1 0 x4
  (x2)2
• (x2)2 ? 2x2 1 0 Let u
  x2
• u2 ? 2u 1 0
• (u ? 1)(u ? 1) 0 Solve the
  quadratic eqn.
• u ? 1 0
  Zero-factor Property
• u 1
• To find x, replace u with x2. x2
  1
• 
  x
• 
  x ?1
• Checking in the original problem, the solution
  set is ?1.

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Another Example

• Solve (x 1)2/3 ? (x 1)1/3 ? 2 0.
• Solution Since (x 1)2/3 (x 1)1/32, let u
  (x 1)1/3.
• u2 ? u ? 2 0
  Substitute.
• (u ? 2)(u 1) 0
  Factor.
• u ? 2 0 or u
  1 0
• u 2 or
  u ?1
• Now replace u with (x 1)1/3
• (x 1)1/3 2 or (x
  1)1/3 ?1
• (x 1)1/33 23 or (x
  1)1/33 (?1)3 Cube each side.
• x 1 8 or x
  1 ?1
• x 7 or
  x ?2

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Another Example continued

• Check (x 1)2/3 ? (x 1)1/3 ? 2 0
• Both check, so the solution set is ?2, 7

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Homework

• 1.6 page 144
• 7 13 odd, 15 25 odd, 27 39 odd, 49 53
  odd, 59, 61 65 odd, 69, 75