254 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

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1 Chapter 9 Radical Expressions and Equations Section 9.. Introduction to Radical Expressions Practice 9... Since negative so this is not defined using real numbers Since Since (/8) (9/) The radicand is negative so this is not defined using real numbers.. Since (). Since (0.) Since (0.07) Practice Practice The radicand is negative so this is not defined using real numbers.. -. The radicand is negative so this is not defined using real numbers. Practice The radicand is 8. ( + 8 )( - ) ( + 9)( - ) (-8) - 9. ( - )( - ) The radicand is negative so this is not defined using real numbers. Practice 9... a 9., b. c a + b c ( 9. ) + (. ) c c 8. c 9.. a 08, b 700 c c, 0, c rounded. s gd g, d s s 0 s 8. g and d 9 s 9 s s. g 9.8 and d s 9. 8 s 9 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

2 s 7 Exercise Set so 8. ( 9 ) ( 8 ) so 8 9. (0.) 0.0 so (.). so The radicand is negative so this is not defined for real numbers The radicand is negative so this is not defined for real numbers ( - 00 ) ( + ) ( - 0)( + ) 9 8. ( + ) ( - ) ( + ) ( - ) ( + ) ( - ) 8(-0) -80. c a + b a, b 8 c + 8 c 9 c 7. a 0.00, b 0.0 c c c a 7, b 00 c c, 0, c 00. a 0.00, b 0.00 c ( 0. 00) + ( 0. 00) c. 7 c s d d 0 feet s 0 s 0 s 90 mph. d 80 feet s 80 s 00 s 0 0 mph Data #7 y x a(a - ab + b) +b(a - ab + b) a - ab + ab + ab - ab + b a + b CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

3 0. x + x h 7h 9 h 7h x - x - x + 0 x + 0 x - h 7 h 7 h x x 7 Subtract - x + 0 x h + x - x Subtract h x - x - Subtract. 8. 9s - t 8x 9 y y x 9 x 9 y y 9. - x + x x x y 0x 0x - x + x 0x - + 0x + 0x 0x 0. - x + x x - x + (x - 9)(x - ). x - 0x + (x - x + ) (x - )(x - ). x + x + 9 (x+)(x+) (x +). 0x + 0x - 7x - 0x - x - (x - x - 7) (x + )(x - 7). ( ( a ) ( ab ) 0a ) ( a b 0a ) ( b a ) b a. abab- ab. ( t v w ) ( t w ) t v w.. x y x y x y y x xy v tw m ( m ) m ( m ) ( m ) ( m ) m m 8 m 9 m 7 m m 7. (s +t)(s - t) (s) - (t). (x -) (x-9) (x-) (x-9) 8x - 8 (x - 9) 8x - 8 x - 8 x 0 x 0. x - 0 (x +8)(x - 8) 0 x or x x -8 or x 8. x - 9 (x+)(x-) LCD x - x + x 9 x - x + ( x + )( x - ) ( x + )( x - ) x - ( x + )( x - ) x + ( x + ) ( x - ) ( x + ) ( x - ) (x+)-(x-) x + - x + 9 -x + -x - CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

4 x /0 - / /0 Undefined is a restricted value. There is no solution. 7. Eq : x - y - Eq : 7x + y times Eq : (x - y) (-) times Eq : (7x + y) Eq transformed: x - y -0 Eq transformed: 77x +y7 Add together 8x x Substitute to solve for y: - y - - y - -y - y The solution is the ordered pair (,). 8. i prt i Total $7,000 + $, $0, 9. 8 right questions x right 8 questions x 8 8 x 8 x 8.0 or 8 questions 70. Let x number of years since 98 y current gold medal distance y x CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS 7

5 . 8w g 8wg w w g g g Practice Practice Section 9. Simplifying Radical Expressions Practice 9... w w w. 8xy 8 xy 9 xy. t t t Practice 9... d d 8 since 8. y t z y t z y t z. x x x since with remainder.. ( a ) a if a 0 Practice 9... d d d d. 8 8 b b b. w h w wh w h. y 8 m y 0 m y since 8 0 with remainder and. x y z 9 x y z xz since with remainder, and 9 with remainder.. m 9 n r m n r mr since 9 with remainder, and with remainder. 8 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

6 Practice a b a b a b a b. 7x 8 y 0 x 8 y 0 x y. 8r s t 7 r s t r 7 s rst 8 r 7 s rst. x 7 y z 8 x 7 y z 8 x 8 y z xy. x x + ( x - ) x - 9. x x x. 9x 9 x x. 7 7 t 7 t 7 t. rw w rw w r r x + x + ( x + ) x Practice m 9m 7 7 m 7 m 7 7 m 7 m m 9m m. b 8b b b b b b 8 b b. x 7x y x x y 8 y y x xy xy 9 x y xy y x x 7 x. a 7 a 8 a. m m 7 n m m 7 n n n m m n m m m n n Exercise Set 9.. 9m 9 m 7 m.. s s s 0 y 7. ( y ) y 0 y 0 y. x 7 y 9 z x y z xyz 7. r s t r s 7 t 8 9. y y y y. 0a 9 b a 9 b a b 7 ab. 00w t 0 w t 0w t 8 w. 80r s t r s t rs t st rs t st CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS 9

7 7. 0x 0 y z x 0 y z x 0 y z z 9. x + x + ( x + 8 ) x + 8. x 8 x + ( x - ) x -. x + 0x + ( x + ) x +. x x 0 x x 0x 7. 8 s s 8 s s s 8s s 9. 9 wx w 9 x 9 wx w x w 08w x w. x y x y x y x y 8 x x 8 y x. 0a 7a b 0 a a b 0 b b a a 8 b a b b 9. y -x y - x m - and the y intercept is (0,) x - (9x+)(9x-) 7. (0,-) and (,0) 7. Data #8 y x x - 0 0x x - 7 ( x - ) ( x x - ) ( x - ) ( x + 9 )( x - ) ( x + 9 ). 8 x 0x y 7 8 x x y 7 y y 8 x x y y 8 x y y y 7. Let r speed at long play r speed at standard distance rate time (r) +.(r) r + r r r. meters per hour 8. Let t time of slow train t - time of faster train Distances are equal. distance rate time 8t + (t -) 8t + t - t 08 t.7 hours 0am +.7 hrs : pm Data #9 y x CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

8 Data #0 y x Eq : x + 8y 0 times Eq : (x + 7y) (-) - times Eq : -(x + 8y) -(0) Eq transformed: x + y - Eq transformed: -x - y -0 Add together y - y -7 Substitute to solve for x: x+7(-7) - x x x The solution is the ordered pair (,-7). 7. x - x - x - x - x - x - -7x - x 80. -(x+) -(8-x) -x x -x x - 0 (x - 9) 0 (x+)(x-) 0 x + 0 or x - 0 x - or x 77. LCD x(x + ) x + + x x + x ( x + ) x + + x ( x + ) x x ( x + ) x + x + x + x x + x x x 78. Eq : x + y Eq : x - y Eq divided by : x - y Substitute Eq in Eq : -y + y This is an identity. There are an infinite number of solutions. Solve for y: y -x The solutions may be written (x,-x). 79. Eq : x + 7y m-(-m)(-m-) 8. ( a b ) ( a b ) ( a b ) a 9 b a 8 b a - b a 8.. x x 0 8b a b 8 8. (x - p)(x + p) x - p 8. ( m n + ) + ( m n - mn+ ) + ( mn- ) m n + mn+ m n + mn+ 87. x + x + 0 (x + x + 0) (x + )(x + ) 88. 8x - 0x + 8x 8x(x - x + ) 8x(x-)(x-) 89. a)let x number of years since CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

9 9 y the winning time y x b) x y (7) y. seconds c) x <0 9. <0.x x >8.8 x > 9 years after 9 Occurred in x 7 y x y 7 x x 7 y y x 7 x 7 y. 8 x y x y x y x y x y 8 x y x y. m n 7 7m n m n 7 m n m n 9 m 8 n 9 n m 8 n 9 n Practice 9... ( ) ( ) Section 9. Multiplying and Dividing Radicals Practice a 7 a 7 a 7 a 77a ( 7 a ) 77a 7 a m m m m m m m m m m ( m ) m m m CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

10 . y 8 y y 8 y y y y y y y y y y y. m ( m ) m m t t t t t ( t ) t t Exercise Set x 0x 9 0x 0x 9 0 x 0 0x 7. 8 t t q q 8 t 8 t q q 8 t 8 t q t 9. m m m m m. x y 7x 0 y x y x y x y 7 y 7x y 7 y. x y x y x y 9 x y 9 x y y 8x y y ( ) 8 7 m 7 m m m. 7. x 0x x 0x 0x 0x x 0x x 0x x 0x ( 0x ) 0x 7 d d 7 d d d d 7 d d 7 d d 7 d d ( d ) d 9. (x-xy) -y x - xy - y (-) - (-)(-) - (-) Eq : 9x + y Eq : x - y Solve Eq for y: y - 9x Substitute in Eq : x - ( - 9x) x - 8-8x x x Substitute to solve for y: 9() + y 9 + y y - The solution is the ordered pair (,-).. Let t time of freight train t - / time of passenger train distance rate time The distances are equal. 0t (t - /) 0t t - / -t -/ -0t - t.7 hours 8: + hr and min pm No, it will not have caught up.. amount base rate Let x percent of copper in gram CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

11 bar. 0(0.0) + x (0+)0. +x x 9.7 x.7 x 0. or %. d / n(n - ) / (n - n) 8 n - n n - n (n - 7)(n + ) 0 n or n + 0 n 7 sides or n - disregard We can t have a - sides.. f(x) x - x + 9 f(-) (-) - (-) + 9 f(-) w w + w + - w + w + w + w 7 w + w + w + w Subtract w 7 w w 8 w Subtract w + w w + w Subtract w + w + 8 Subtract -. (0z-y)(0z+y) 00z - y 7. -p( t p ) ( p t ) - p ( t - 0t p ) ( p t ) 8. (xy)(-xy)- xy(-)-x-y- (-)-xy 7xy 9. (d-)(d +7) d +8d - d - d + d - 0. ( m )( m ) ( m ) n n n ( m n ( m n ( m n m n 7. ( b b ). ( - ) b b - 8 b 9 m m - m )( ( ) m )( m ) n n )( m n ( - ) )( m ) n )( m n )( m ) n b ( - ) b m ( ) m m ( ) m - m 0. m n m n m n mn m n mn. 8x - 0x - (x - x - ) (x + )(x - ). 9x - (x + )(x - ). x + x + x - 0x 8x + x - 0x 8x(x + x - ) 8x(x + )(x - ) 7. x - x (x - )(x + ) 0 x - 0 or x + 0 x or x - 8. x + x - x + x + 0 (x + x + ) 0 (x + )(x + ) 0 x + 0 or x + 0 x - or x - x -/ or x - 9. x - x - (x - 7)(x + ) LCD 7 x x - + x + x - 7 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

12 7 ( x + )( x - 7 ) + x + x ( x + )( x - 7 ) ( x + )( x - 7 ) + ( x + )( x - 7 ) x + ( x + ) ( x - 7 ) x (x -7) (x + ) 7 + x - x + + x x + This is a contradiction. There is no solution. 0. a) 80 years after 89 (99) there were 0.7 million students enrolled. b)this represents the number of students enrolled 00 years after 89 (99). c) t years after 89 there were 0 million students enrolled. d)the number of students enrolled t years after 89 is times the number of students enrolled in 89. Section 9. Adding and Subtracting Radical Expressions Practice y 8 y y This cannot be simplified any further.. t 8 t - 7 t. y + 8 y 7 y y 7. 7 m + 7 m 7 m 7 m + 7 m 8. 8 s 8 s 8 s s Practice CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

13 m 0m + 8 m m m + m m m + m m 0m m + m m m m. 9t + t t 7 t + 8 t t t + 8 t t t Practice 9... ( 0 + ) y ( y + ) - y 7 y + 8 y y 7 y + y. t ( t + ) - t t + 9 t t t + t. ( x + ) + x ( x ) + x + ( ) + x x + x x x + 8 x ( ) Exercise Set s + 9 s s s 7. + x + 7 x 0 x 9. a a. c c c c 8 c a a - a. n + 8 n n + n 9 n. t - w This cannot be simplified. 7. s s s s s y + 9y 8 y CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

14 y + 7 y 8 y y + y 0 y - y x x + 7x x x + x x x + x x 0x x + 0x x 0x x 9. r r r r r r r 9 r r r r r r r r r r r r r r r - r r. ( 8 + ) ( + ) ( ) w ( w + ) + w w + w + w w + 0 w 9. ( a ) + a ( a ) a a a a a a - a + 9. ( w ) w ( w ) w + ( - ) w w - w + - w ( ) ( ) ( ) The base is and the exponent is Let x the first integer; x + second integer; x + third integer x + (x + ) + (x + ) (x + ) +9 x + x+ +x + x x + x + x 0 x x CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS 7

15 . y x - m Slope of a perpendicular line is -/. y mx + b 0 -/(0) + b b 0 y -/x. Percent for math and computer sciences,000,000/,9,000,000 x00%.7% Total in 990 9,79,000,000 Percent for math and computer sciences 8,000,000/8,88,000,000x 00% 9.% The percent for math increased.. -x + x + -(x -7x - ) -(x + )(x - ). x - 0x 0 ( x - ) 0( x ) ( x ) ( x + )( x - ) ( x + ) 7. 8( - ) 8( - 9) 8(- 8) 8(7) 8. - x + x ( x x ) - x + x ( ) x - + x + ( x + 7 ) x x x x + 7 x x x - x x - x + x + ( x + ) ( x - ) x - ( x - ) ( x + ) ( x + ) ( x + ) x - 8x y 9 x x y 9 x xy 9 x 9 x x. ( 9 ) ( 8 - ) ( ). a 9 b a ( b ) ( b ) a b a. -(f-g-h)-(h-f)-(-g+h) -f+g+h-h+f+g-h g - h. 0 x - x + 0 (x - 8)(x - ) x or x - 0 x 8 or x. -0 -x 0x 0x + x ac Product Sum 0() () 0x + x + x x(x + ) + (x + ) 0 (x + )(x + ) 0 x + 0 or x + 0 x - or x - x -/ or x -/. 0.(-x)+0.x+0.x.8-0.x + 0.x +0.x x +0.x.8 This is a contradiction. There is no solution m m - m m 7 m m m m - m 8 m -(m-) 8 m - m m m The domain is -7 x. The range is y a)e t b)e(0) (0) 79.9 By 00 (90 + 0) the average life expectancy will be about 79.9 years. c) t t 8 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

16 t -.7 years This means in about 89 (90 - ) the life expectancy was 0 years. d)e(0) 70. (70.) t t t 9.7 years In 99 the average life expectancy will be twice the average life expectancy in 90. Section 9. Solving Equations that Involve Square Roots Practice 9... x 0 ( x ) ( 0) x 00 x 0 Check: x ( ) ( x ) 9(x) x x Check: ( ). x + ( x + ) ( ) CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS 9

17 x + 9 x Check: + 9. x - 0 ( ) ( x - 0 ) x - 0 x Check: x Practice 9... x - x 0 ( x ) ( 0) x 00 x 0 Check: x - x - ( x ) ( - ) x x Check: - ( ) x. + - x - x - x ( - x ) ( ) -x -x 9 x -9 Check: -( - 9) x + x + 9 9(x+) x + 9x + x + 8 x x Check: + 9 9/ 8 +. x - 7 x + ( x - 7 ) ( x + ) (x-7) x + x - 7 x + x 7 x Practice x x 0 - x x Square both sides x + x (x + )(x -) 0 x + 0 or x - 0 x - or x Check x -: 0 - ( - ) is not a solution. Check x : 0 - is a solution.. x + - x+ Square both sides. x x Check: ( ) + - is not a solution. There is no solution. x +. x + ( x + ) x + ( ). x x 9 x Square both sides. x. Check: (. ) CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

18 is not a solution. There is no solution.. x x + 8 x x+8 Square both sides. x - x - 80 (x -x -) 0 (x -)(x+) 0 x - 0 or x + 0 x or x - Check x: () ( ) + 8 is a solution. Check x -: (-) ( - ) is not a solution. Practice 9... d. h h + 8 feet d. ( 8) d 7 d. miles. d. h.. h.8.h Square both sides. h. feet. d. h 8. h.h Square both sides. h.7 feet feet for the platform. Practice 9... v µgr v 0. 0( 9. 8 ) 00 v 9 v meters per second. v µgr µ 078µ µ 0. Exercise Set 9.. x 0 x 00 Square both sides. x 0 Check is left to student.. w w Square both sides. w 8. x x Square both sides. 0x x. 7. t t 7 8t 7 t 9. x + x + x x Square both sides.. x - 9 x - Square both sides. x x 7. y + 7 y + 9 y. x x x - x - 0 This is a contradiction. There is no solution x 7 x 7 7x 9 Square both sides. x x + x x Square both sides. x. x 7 - CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS 7

19 x x Square both sides. x. x 7 x x x. x 0 - x x Square both sides x x - x x Square both sides. x x x x x + x. + x 0 + x 00 + x Square both sides. 9 x x x x - 8 x - 8x Square both sides. -8x x - - x - - x - x - x Square both sides. -x 0 x - 7. x - 0 x + (x -0) x + Square both sides. 7x - 00 x + x x x - x + (0x-) x + 0x - x + x 0 x. y + 7 y y + 7 y Square both sides. y - y (y -9)(y + 8) 0 y or y y 9 or y -8 Check: y y 9 is a solution. Check y -8: is not a solution.. - x 7 - x x Square both sides. x Check : is not a solution.there is no solution.. x 0 - x x 0 -x Square both sides. x + x (x + )(x - ) 0 x + 0 or x - 0 x - or x Check x -: - 0 -( - ) is not a solution. Check x : 0 - is a solution. 7 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

20 7. - x - 0 -x 00 Square both sides. -x 9 x Check: - ( ) There is no solution. 9. x 0x + x 0x + Square both sides. x - 0x - 0 (x - x - ) 0 (x - )(x + ) 0 x - 0 or x + 0 x or x - Check x : () 0 + is a solution. Check x -: (-) 0( - ) is not a solution.. p p + 8 p p +8 Square both sides. p -p -8 0 (p- )(p+) 0 p - 0 or p + 0 p or p - Check p : ( ) + 8 ; is a solution. Check p -: - ( - ) ; - is not a solution.. x + x x Square both sides. x Check: is not a solution.. h + feet d. ( ). 7 miles 7.. h.h Square both sides. 0 feet h 0 - feet taller 9. d miles d.. 0. miles miles farther. () + x (0) 0 feet + Joline x Scott feet x (x + 90)(x - 90) 0 x or x x -90 or x is not valid so x 90 feet is the height of the kite.. m y mx + b y x - undefined.. m - - ( - 8 ) 0 0 Since m 0, the equation is y. x - 7x + 8 (x -)(x - ). x - 9x - 8 (x + )(x - ) x CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS 7

21 7. xyz xyz xyz xyz 0xyz xyz GCF xyz xyz 8. 8x - 8x 0 8x(x-) 0 8x 0 or x - 0 x 0 or x 9. Eq : x + y Eq : x + y - - times Eq : -x -y - Add together 0-9 This is a contradiction. There is no solution b + b + b + ( b + ) + ( b + ) b + b + b + ( b + ) b + + b + b + b + b - Check: /0 /0 This is undefined. There is no solution ( - ) - + ( - ) + (-) s s s 7 9 s s s 7 9 s s s 7 s 7 s The x intercept is (-8,0) and the y- intercept is (0,). 78. Yes, it passes the vertical line test. 7. x - x x(x - ) x - (x +)(x -) LCD x(x - )(x + ) x x + x x ( x - ) + ( x + )( x - ) x ( x - ) x + x + + x x ( x + )( x - ) ( x + ) + x x ( x + )( x - ) x + x ( x + )( x - ) x + + x x ( x + ) ( x - ) 7. ( xy ) ( x y ) ( xy ) x y x y x y x y (x-) - ( - x) 8 - x x 9 + x 7 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

22 Section 9. Quadratic Equations Practice 9... x + 7x - 0 a, b 7, c - x 7 ± 7 ( )( - ) ( ) 7 ± x x 7 ± x 7 ± Solution Solution x (-7+)/ x (-7 -)/ x / x -8/ x / x -. x - x + 0 a, b -, c x ( - ) ± ( - ) ( )( ) ( ) x ± ± Solution Solution x +. -x + x a -, b, c 7 x ± ( ) ( - )( 7 ) ( - ) x ± ± 7 x - Solution Solution x + 7 x - 7 x - 7..x - x -. 0 a., b -, c -. x + 7 x ( - ) ± ( - ) (. )( -. ) (. ) ± +. ±. x.. Solution Solution x x.. x x x.89 x -.80 Practice 9... x x - x - x - 0 a, b -, c -. x ( - ) ± ( - ) ( )( - ) x ± + 8 ± ± x ± ± Solution Solution x + x - x x 0 x x - a /, b -, c -/ x ( - ) ± ( - ) ( ) ( - ) ( ) x ± x ± ± 9 Solution Solution 8 8 x + x -. - x - x x - x + 0 a, b -, c x ( - ) ± ( - ) ( ) ( ) ( ) 7 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

23 x ± ± 8 8 There is no real solution. The radicand is negative.. 0+ x 0.x 0.x - x a 0., b -, c -0. x ( - ) ± ( - ) ( 0. )( - 0) ( 0. ) x ± + 8 ± ± 0± 0 0. Solution Solution x x x x - 9 x x + 0 a /9, b -/, c x ( ) ± ( ) ( 9 ) ( ) ( 9 ) x ± x 9 ± 0 Practice 9... a)y -x +0x + 7 y -() + 0() + 7 y or 9 feet below ground level b)0 -x + 0x + 7 -x + 0x - 0 a -, b 0, c- x 0± ( 0) ( - ) ( - ) ( - ) 0± 00 x 0± 88 0± 8. 7 x Solution Solution x x x0.7 seconds x.08 9 seconds c) When the ball hits the ground the height is x + 0x + 7 a -, b 0, c 7 x 0± ( 0) ( - ) ( 7 ) ( - ) 0± x 0±. x Solution Solution x 0 +. x-0. (Invalid) Exercise Set 9.. x - x - 0 a, b -, c - x 0 -. x.8 sec. x ( - ) ± ( - ) ( )( - ) ( ) x ± + 0 ± ± 8 Solution Solution x -/ - x 0/ /. 0 x + x + a, b, c x ± x ± ± 9 x ± Solution Solution x -8/ - x -/ -/. x - 9x + 0 a, b -9, c x ( - 9 ) ± ( - 9 ) ( )( ) ( ) 9 ± 8 x 9 ± Solution Solution x 9 + x 9 - CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS 7

24 7. -x +x+0 a -, b, c x ± ( - ) ( ) ( - ) x ± + 8 ± 8 Solution Solution x + x 8 8 x x x - 9x - x + 0 a 9, b -, c x + 8 x ( - ) ± ( - ) ( 9 )( ) ( 9 ) ± x ± x 8. x + x - 0 a, b, c -/ x ± ( ) ( ( ) x ± 9 + x ± 7 ) ± x - ± Solution Solution x - + x x +0.0x x -.07x -. 0 a 0.0, b -.07, c -. x (-. 07) ± (-. 07) ( 0. 0) (-. ) ( 0. 0). 07± x 0. x. 07± ± Solution Solution x x x x.98. x + x - x +x + 0 a, b, c x ± ( ) ( )( ) ( ) x ± ± 8 8 The radicand is negative. There is no real solution. 7. -x +0.x x - x a 0., b -, c 0. x ( - ) ± ( - ) ( 0. ) ( 0. ) ( 0. ) x ± ± 0 9. a)y 0.x +.x ,000 0.x +.x x +.x -. a 0., b., c -. x. ± (. ) ( 0. ) (-. ) ( 0. ). ± x ± 00. x. ± x 9. years; x -.9 years; We will disregard this value since our model is not valid for years prior to 98. b) y 0.(90) y 0.(800) y y thousand births c) y 0.() +.() y y y 07. thousand births in CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

25 7. 0.x +.x x +.x a 0., b., c -9.7 x. ± (. ) ( 0. ) (- 9. 7) ( 0. ). ± x ± ± x x. years; Again we disregard the negative values for x, since our model is not valid for years before 98.. amount rate time amount in amount out Let x time to fill 7x 0(x +.) 7x 0x + 90 x 90 x hours 7 0 gallons. < >. x - (x + )(x + ). (st)-(s-t-) s-t-s -t- s-8t- s 8 t x0 ( 0. 0) ( ). 08 x0. x 0 ( x 0 ). 08 x0. 8 x x 0. x0 x 0. x0 y x z x 0y z 9 y x z x y z x z y x + x x - x ( x + ) x - x x + x ( x + ) ( x - ) x ( x + ) ( x + ) Data # y n n - 7 n 7 n n + 7 n 7 n Subtract 0 - n + 7 n + 7 Subtract x x x x ( x - ) x + x x ( x x ) x ( x + ) x - x ( x - )( x + ) x ( x + ) x - x x - x - x - x - - ( x - ) - x + - x x - x - x - + t t t ( + ) t t ( ) + t + t t.. ab 9 b a.. 8x 7 y x 7 y x y x CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS 77

26 (-) -- (-) -- (-) x + x - 0 (x - )(x + ) 0 x - 0 or x + 0 x or x - x / or x - 9. z - + z - z - z - Square both sides z 0 Check is left to student. 0. Eq : x - y Eq : -x+ 8y times Eq : x - 8y Add together 0 9 This is a contradiction. There is no solution.. Yes, it passed the vertical line test. 78 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

27 x y z xy z z. f h 9 fh fh. 08s t s t s t s t. 700m n p 7 00m n p 0mn p 7 mp Chapter 9 Review Exercises. 9 The radicand is negative. Not a real number.. -. ( 9 ) ( 7 - ) ( ). t 7 s 7 7. t 9s 7 s s t 7 s s t s 7 s s 8. b b cd 8 b b cd 8 b b d bc 8 b d bc 9. a a b 8a 7 b a b a b 9 a 7 b a b a a a b a b. 9x y 7 xy. 9 m n m n 0. x 0x y y 0x y 8 x 0x y y x y 8 x y 7. 0m m 00m 00m 0m m x x x x y 8y y 8y y y CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS 79

28 y y y y y y y. ( x + x ) ( x ) + ( x ) ( x ) + x x + x x + x. t + t t t 9 t t t. r r + r r r + r. x + 7x x 7x x + x x x x x + x x x x x x t ( t t ) + ( t + t t ) t - t t + t + t t t 9. y - y - Square both sides. y 0 y 0 0. t t t t t Square both sides.. t + t + 8 -t + 9(t +8) Square both sides. -t + 8t + 7 -t t - Check: ( - ) + ( - ) is a solution x x - - x -x x ( - 7 ) is a solution.. x + - x x x ( ) + + This is a contadiction. There is no solution.. x + + x x + x - x + (x -) Square both sides. x + x - 0x x x This is a contradiction. There is no solution.. -x -x + 0 a -, b -, c x ( - ) ± ( - ) ( - )( ) ( - ) x ± 9 + ± ± Solution Solution x + x - x - x /. y -8y- 0 a, b -8, c - y ( - 8 ) ± ( - 8 ) ( ) ( - ) ( ) 80 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

29 y + y - 8 ± + 8 y 8 ± 8 8 y 8 ± 7 7 ± 8 Solution Solution q - q a, b -/, c /9 q - d 00 feet ± ( q ± 9 9 q ) ( ) ( 9 ) ( ) ± 0 00 feet (00) + (00) d d d a, b 0, c d 0 ± 0 ( )( ) ( ) d 0 ± ± 7. d. The negative value is invalid since we can t have a negative distance more feet extra seconds to walk the edge. 8. b b+.7 b - b a, b -, c -.7 b ( - ) ± ( - ) ( ) ( -. 7) ( ) b ± + 7 b ± ± 8 ± Solution Solution b + b 9. distance rate time distance/rate time CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS 8

30 - ( + ) - 8 ( 0 + ) - 8. t a t a. 0y y y y y y. t + t t + 80t t t + t 9 t + t t t t + t t + t t t t h j 7 h j hj 7. ( 0 ) - ( + 0 ) n + n + ( n + ) n + 9. a b 0b c c a b c ab c a b c Chapter 9 Test. - ( + ) 0. y + 7 y y + y + y + y + y y + y y 8 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

31 . y y y 7 y 7 y y 7. d d d d d. x 9(x) Square both sides 8x x 8 Check left for student.. x + x + x x + x + x + x + x + (x + ) Square both sides. x +x+ feet 7 feet (x + )(x -) x + 0 or x - 0 x - or x Check x -: ( - ) + ( - ) + ( - ) ; - is not a solution. Check x ( ) + ( ) + ( ) 9 - ; is a solution. x +x+ 0 x - 0. x + x - 9(x+) x - Square both sides. 9x + 9 x - 7x - x - Check: - + ( ) - 9 The radicand is negative. There is no real solution. d. x +x+ 0 a, b, c x ± ( ) ( )( ) ( ) ± 9 ± x x ± Solution Solution x + x - x -8/ -/ x -8/ -/ 7. y - y - 0 a, b -, c - 8. y ( - ) ± ( - ) ( ) ( - ) ( ) y ± 9 + ± + 7 d d + 9 d d - 0 a, b 0, c - d 0 ± 0 - ( )( - ) ( ) d ± 0 ±. d 8.0 feet (Disregard the negative value. It is invalid.) CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS 8

32 8 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS