47. Mental Math What is the solution? x + 11 = 4 square root of x plus 11 end root , equals 4
48. You can find the area A of a square whose side is s units with the formula A = s 2 . eh equals , s squared , .  What is the best estimate for the side of a square with an area of 32 m 2 ? 32 , m squared , question mark
    1. 4.2 m
    2. 5.7 m
    3. 8.0 m
    4. 16 m

Solve. Check for extraneous solutions.

49. 3 2 x − 3 = 9 3 , square root of 2 x end root , minus 3 equals 9
50. 2 ( 2 x ) 1 3 + 1 = 5 2 . open , 2 x , close super 1 third end super . plus 1 equals 5
51. 2 x − 1 − 3 = 0 square root of 2 x minus 1 end root . minus 3 equals 0
52. ( 2 x + 3 ) 1 2 − 7 = 0 open , 2 x plus 3 , close super 1 half end super . minus 7 equals 0
53. x 2 + 3 = x + 1 square root of x squared , plus 3 end root . equals x plus 1
54. ( 2 x + 3 ) 3 4 − 3 = 5 open , 2 x plus 3 , close super 3 fourths end super . minus 3 equals 5
55. 2 ( x − 1 ) 4 3 + 4 = 36 2 . open , x minus 1 , close super 4 thirds end super . plus 4 equals 36
56. x 1 2 − ( x − 5 ) 1 2 = 2 x super 1 half end super , minus . open , x minus 5 , close super 1 half end super . equals 2
57. x = x − 8 + 2 square root of x equals , square root of x minus 8 end root , plus 2
58. ( x − 3 ) 2 3 = x − 7 open , x minus 3 , close super 2 thirds end super . equals x minus 7

59. Error Analysis A student said that 4 and 1 are the solutions of the problem shown. Describe and correct the student's error.

    

60. Physics The velocity v of an object dropped from a tall building is given by the formula v = 64 d , v equals , square root of 64 d end root , comma  where d is the distance the object has dropped. Solve the formula for d.
61. Open-Ended Write an equation that has two radical expressions and no real roots.
62. Reasoning You have solved equations containing square roots by squaring each side. You were using the property that if a = b eh equals b  then a 2 = b 2 . eh squared , equals , b squared , .  Show that the following statements are not true for all real numbers.
    1. If a 2 = b 2 eh squared , equals , b squared  then a = b . eh equals b .
    2. If a ≤ b eh less than or equal to b  then a 2 ≤ b 2 . eh squared , less than or equal to , b squared , .
63. A teacher asked students why it is necessary to check for extraneous roots when squaring both sides of the equation. Which of the following answers is the best? Is this answer complete? Explain.
    1. Because the squared equation can have negative roots.
    2. Because squaring is multiplication, and any multiplication is a potential source of extraneous roots.
    3. Because when you square both sides of the equation a = b , eh equals b comma  you add to the solution set the roots of the equation a = − b . eh equals negative b .
    4. Because any operation with an equation may result in extraneous roots.

Solve. Check for extraneous solutions.

64. x + 1 + 2 x = 5 x + 3 square root of x plus 1 end root , plus , square root of 2 x end root , equals . square root of 5 x plus 3 end root
65. x + 2 x = 2 x square root of x plus , square root of 2 x end root end root . equals , square root of 2 x end root
66. x + 2 x = 2 square root of x plus , square root of 2 x end root end root . equals 2
67. x + 25 = x + 5 square root of square root of x plus 25 end root end root . equals , square root of x plus 5 end root

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