Introduction to Radical Notation

Chapter
8
Quadratic Functions
8.1
Introduction to Radical Notation
1. We are looking for a number whose square is −400.
• However, every time you square a real number, the result is never negative. Hence, −400 has no real square roots.
3. We are looking for a number whose square is −25.
• However, every time you square a real number, the result is never negative. Hence, −25 has no real square roots.
5. We are looking for a number whose square is 49.
• Because (−7)2 = 49, the negative square root of 49 is −7.
• Because (7)2 = 49, the nonnegative square root of 49 is 7.
Hence, 49 has two real square roots, −7 and 7.
7. We are looking for a number whose square is 324.
• Because (−18)2 = 324, the negative square root of 324 is −18.
• Because (18)2 = 324, the nonnegative square root of 324 is 18.
Hence, 324 has two real square roots, −18 and 18.
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CHAPTER 8. QUADRATIC FUNCTIONS
466
9. We are looking for a number whose square is −225.
• However, every time you square a real number, the result is never negative. Hence, −225 has no real square roots.
11. Every time you square a real number, the result is never negative. Hence,
the equation x2 = −225 has no real solutions.
13. There are two numbers whose square equals 361, namely −19 and 19.
x2 = 361
x = ±19
Original equation.
Two answers: (−19)2 = 361 and (19)2 = 361.
Thus, the real solutions of x2 = 361 are x = −19 or x = 19. Writing x = ±19
(“x equals plus or minus 19) is a shortcut for writing x = −19 or x = 19.
15. Every time you square a real number, the result is never negative. Hence,
the equation x2 = −400 has no real solutions.
17. There are two numbers whose square equals 169, namely −13 and 13.
x2 = 169
x = ±13
Original equation.
Two answers: (−13)2 = 169 and (13)2 = 169.
Thus, the real solutions of x2 = 169 are x = −13 or x = 13. Writing x = ±13
(“x equals plus or minus 13) is a shortcut for writing x = −13 or x = 13.
19. There are two numbers whose square equals 625, namely −25 and 25.
x2 = 625
x = ±25
Original equation.
Two answers: (−25)2 = 625 and (25)2 = 625.
Thus, the real solutions of x2 = 625 are x = −25 or x = 25. Writing x = ±25
(“x equals plus or minus 25) is a shortcut for writing x = −25 or x = 25.
√
21. The expression 64 calls for the nonnegative square root of 64. Because
(8)2 = 64 and 8 is nonnegative,
√
64 = 8.
Second Edition: 2012-2013
8.1. INTRODUCTION TO RADICAL NOTATION
467
√
23. The expression − −256 calls for the negative square root of −256.
√ Because you cannot square a real number and get −256, the expression − −256
is not a real number.
√
25. The expression − 361 calls for the negative square root of 361. Because
(−19)2 = 361 and −19 is negative,
√
− 361 = −19.
√
27. The expression − 100 calls for the negative square root of 100. Because
(−10)2 = 100 and −10 is negative,
√
− 100 = −10.
√
29. The expression 441 calls for the nonnegative square root of 441. Because
(21)2 = 441 and 21 is nonnegative,
√
441 = 21.
√
31. If a > 0,√ then − a is the negative solution of x2 = a. Hence, when we
substitute − a into the equation x2 = a, we must get a true statement:
√
(− a)2 = a.
Thus:
√
(− 17)2 = 17.
√
33. If a > √
0, then a is the nonnegative solution of x2 = a. Hence, when we
substitute a into the equation x2 = a, we must get a true statement:
√
( a)2 = a.
Thus:
√
( 59)2 = 59.
√
35. If a > 0,√ then − a is the negative solution of x2 = a. Hence, when we
substitute − a into the equation x2 = a, we must get a true statement:
√
(− a)2 = a.
Thus:
√
(− 29)2 = 29.
Second Edition: 2012-2013
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CHAPTER 8. QUADRATIC FUNCTIONS
√
37. If a > √
0, then a is the nonnegative solution of x2 = a. Hence, when we
substitute a into the equation x2 = a, we must get a true statement:
√
( a)2 = a.
Thus:
√
( 79)2 = 79.
39. Enter each side of the equation x2 = 37 in the Y= menu, then adjust
the WINDOW parameters so the intersection points are visible in the viewing
window.
Use the 5:intersect utility on the CALC menu to find the points of intersection.
Reporting the solution on your homework: Duplicate the image in your
calculator’s viewing window on your homework page.
Second Edition: 2012-2013
8.1. INTRODUCTION TO RADICAL NOTATION
y
469
y = x2
50
y = 37
−10
−6.082763
10
x
6.082763
−50
Next, solve the equation algebraically, then enter the results in your calculator to see if they match those found above.
x2 = 37
√
x = ± 37
Note how these solutions match those found using the 5:intersect utility
on the calculator.
41. Enter each side of the equation x2 = 11 in the Y= menu, then adjust
the WINDOW parameters so the intersection points are visible in the viewing
window.
Use the 5:intersect utility on the CALC menu to find the points of intersection.
Second Edition: 2012-2013
CHAPTER 8. QUADRATIC FUNCTIONS
470
Reporting the solution on your homework: Duplicate the image in your
calculator’s viewing window on your homework page.
y
y = x2
30
y = 11
−10
−3.316625 3.316625
10
x
−30
Next, solve the equation algebraically, then enter the results in your calculator to see if they match those found above.
x2 = 11
√
x = ± 11
Note how these solutions match those found using the 5:intersect utility
on the calculator.
Second Edition: 2012-2013