Section.5 - Other Types of Equations The numbers of solutions to a polynomial with n degree, where n is Natural Number, are n solutions. Solving a Polynomial Equation by factoring Eample Solve: = - = ( ) =, Eample Solve: + = 8 + 8 7
Solving a Radical Equation Power Property If P and Q are algebraic epressions, then every solution of the equation P Q is also a solution of the equation P n Q n ; for any positive integer n. Eample Solve 5 5 5 5 5 ( )( 5) 5 5 Check 5 5 () 5 5 ( 5) 9 5 5 ( true) 5 5 ( false) is the only solution 8
Solving Radical Equations of the Form Assume that m and n are positive integers If m is even: m n k n m m n k m n n m k k n m If m is odd: m n k n m m n k n m k n m Eample Solve: a) 5 5 5 5 5 5 5 5 5 5 b) 8 / 8 9
Equations that Are Quadratic in Form a b c a b c a b c u u n n a b c u au b c Eample Solve: 5 6 5 6 or U 5U 6 U 5U 6 Solve for U ( 5) ( 5) ()(6) U () 5 5 5 U 5 U 5 U
Eample / / Solve: u / u u ( u )( u) u u u u u / u / 8 7 / / Eample Solve: / / Or factor / / / / U U U U Solve for U / / U = ( ) ()( ) () 6 / U / / 6 6 6 7
Solving an Absolute Value Equation If c is a positive real number and X represents any algebraic epression, then X = c is equivalent to X = c or X = - c Properties of Absolute Value X = c X = c or X = c. For b >, a b if and only if (iff) a b or a b. a b iff a b or a b For any positive number b:. a biff b a b. a biff a b or a b Eample Solve: = 5 = 5 = -5 = 6 = - = = - s: = -, Eample Solve: - - = - = - = 5 - = 5 - = -5 - = - = -6 = - = s: = -,
Eercise Section.5 - Other Types of Equations. Solve: 8. Solve. Solve: 5. Solve 9 9 5. Solve: 7 89 6. Solve 7. Solve: 8. Solve: 9. Solve:. Solve:. Solve: 5. Solve:. Solve. Solve 5. Solve: 6. Solve 7 5 5 7. Solve 8. Solve / 5 6 9. Solve /. Solve: 5/. Solve 7 5 6. Solve 7. Solve 7 6. Solve equation: 5
5. Solve equation: 5 6. Solve equation: 6 7. Solve equation: 8. Two vertical poles of lengths feet and feet stand 5 feet apart. A cable reaches from the top of one pole to some point on the ground between the poles and then to the top of the other pole. Where should this point be located to use feet of cable? L L 9. Towns A and B are located 6 miles and miles, respectively, from a major epressway. The point on the epressway closet to town A is miles from the point on the epressway closet to town B. Two new roads are to be built from A to the epressway and then to B. a) Epress the combined lengths of the new road in terms of. b) If the combined lengths of the new roads is 5 miles, what distance does represent?
Section.5 - Other Types of Equations Eercise Solve 8 ( 8) ( ) Eercise Solve 5 5 5 5 i 5 Eercise Solve: 5 5 5 8 5
5 5 ()( ) () i i i Eercise Solve 9 9 Assume u u Then we can rewrite the equation in a quadratic form: Solve for u using the quadratic formula. u 9 9 9 9 9 9 8 9 8 u 9 6 8 8 u 9 8 8 Since u u u 9u 9u b b ac a 6
Eercise Solve: 7 89 89 7 89 7 Assume: u u 89u 7 u 89 89 7 89 5 89 7 89 7 u 9 89 7 u 8 9 8 8 6 5 5 Eercise Solve ( )( ) Check ( ) ( ) () ()
9 False True is the only solution Eercise Solve: 6 9 7 6 ( 7) ( 7) () (6) () 7 9 7 5 7 5 75 6 75 =, 6 Check: = 5 = (Not a solution) = 6 6 6 6 = 6 = 6 is the only solution Eercise Solve Square both side a b a ab b
Solve for = 5, Check = 5 5 5 5 6 5 : = 5 = - 9 5 ( False) Eercise Solve: ( ) ( ) 8 8 8 =, 6 Check b b ac ( 8) ( 8) ()() 8 6 8 8 6 8 a () : = () 6 (6) 6 5
Eercise Solve: 5 5 5 5 5 5 = = Check: 5 = (True statement) = is a solution 5
Eercise Solve: ( ) ( )( ) Check () ( ) ( ) ( ) 9 ( true) ( true) and are solutions a b a ab b Eercise Solve: 5 5 5 5 5 5 6
= = Check: 5 = (True statement) = is a solution Eercise Solve: 5 ()( ) Check ( ) ( true) ( true) s:, 7
Eercise Solve let u u u (u )(u) u u u 6 u ( ) ( ) ()() u () 5 5 5 6 5 8 The solution set is 6, Eercise Solve 7 5 5 5 5 5 5 8
Eercise Solve 5 5 U 5U ( 5) 5 ()() Solve for U U () U 5 9 5 5 5 Eercise Solve 5 5 5 5 i 5 Eercise Solve Impossible 6 9
Eercise Solve / 5 6 5 6 / 5 6 5 6 Eercise Solve / / 8 8 ( 7)( ) 7 7 5
Eercise Solve: 5/ /5 Reciprocal or use the calculator: ^(/5) = 5 5 Eercise Solve 7 5 6 7 5 6 7 5 5 7 5 5 5 s: 5 5 5 Eercise Solve 7 6 7 6 7 No solution or, since the absolute value can't be a negative 5
Eercise Solve 7 7 7 Distribute 7 7 7 7 7 s:, 7 Eercise Solve equation: 5 5 5 5 5 5 5 5 5 7 7 7 7 Eercise Solve equation: 5 5 5 7 7 5
Eercise Solve equation: 6 6 6 ( ) 6 ( ) 6 ( ) 6 6 6 6 9 9 9 Eercise Solve equation: ( ) :, 5
Eercise Two vertical poles of lengths feet and feet stand 5 feet apart. A cable reaches from the top of one pole to some point on the ground between the poles and then to the top of the other pole. Where should this point be located to use feet of cable? L L l l l l l 6 5 l 6 5 5 6 5 5 6 5 576 8 6 6 5 576 6 8 6 67 8 6 67 8 6 67 8 6 9 6 789 6 9 6 789 686 9 6 789 686 6 5 Solve for : =.59 Square both sides 5
Eercise Towns A and B are located 6 miles and miles, respectively, from a major epressway. The point on the epressway closet to town A is miles from the point on the epressway closet to town B. Two new roads are to be built from A to the epressway and then to B. a. Epress the combined lengths of the new road in terms of. b. If the combined lengths of the new roads is 5 miles, what distance does represent? d d a) d 6 d 6 d d 9 d d 6 9 6 9 5 b) 6 5 9 6 5 5 6 5 5 5 5 5 9 5 576 66 696 9 6 77 576 66 696 58 76 Solve for : 8 55