Expert: Richard J. Raridon - 1/6/2011

Question
Solve:
n4 - 2n2 - 8 = 0  

2. Solve:
c3 + c2 - 7c - 3 = 0, given root -3  

3. Solve:
y4 - 3y3 - 2y2 + 10y - 12 = 0, given root 1 + i

4. Choose the real solutions of the system:
x2 + 2y2 = 12 and 3x2 - y2 = 8
 
5. Choose the real solution of the system:
9x2 + 9y2 = 1 and x = y2 + 1
 
6. Choose the real solutions of the system:
2x2 - y2 = 7 and xy = 3

7. Choose the real solutions of the system:
x = y2 + -9 and x - 4y - 12 = 0
 
8. Choose the real solutions of the system:
y2 = 2x  and  x2 + y2 = 8
 
9. Choose the real solutions of the system:
xy + 6 = 0 and x - y = -5
 
10. Choose the real solutions of the system:
4x2 - y2 + 12 = 0 and x + y = 3

11. Choose the real solution of the system:
8x2 + y2 = 25 and 8x2 - y2 = 39
 
12. Find the length of the legs of a right triangle having perimeter 56m if the hypotenuse is 25m.
 
13. Find the dimensions of a rectangle that has area of 10 and a diagonal of length 5.

14. Four squares, each with sides 4 cm long, are cut from the corners of a rectangular piece of cardboard having an area 560 cm2.  The flaps are then bent up to form an open box having volume 960 cm3.  Find the dimensions of the original piece of cardboard.

15. A 20 m ladder and a 15 m ladder were leaned against a building.  The bottom of the longer ladder was 7 m farther from the building than the bottom of the shorter ladder, but both ladders are the same distance up the building.  Find the distance.

16. Find the length of a side of each of two squares given that the sum of their perimeters is 44 ft and the sum of their areas is 73 ft2.

17. The perimeter of a rectangular lot is 88 m, and its area is 480 m2.  Find the width and the length.

18. Find the dimensions of a rectangle with an area of 60 ft2 and a diagonal of 13 ft.

19. The area of a rectangle is 48 m2. The length of a diagonal is 10 m. Find the perimeter of the rectangle.

20. Find two negative numbers such that the sum of their squares is 170, and twice the square of the first minus three times the square of the second is 95.

21. A number y varies jointly as x and one over the square root of z.  If y=300 and x=5 and z=9, what is y when x=18 and z=36?

22. A number y varies jointly as x and the cube of z. If y = 160 when x = 4 and z = 2, what is y when x = -5 and z = 3?

23. A number y varies directly as x and z and inversely as the square of r. If y = 6 when x = 3, z = 4, and r = 7, wheat is y when x = 6, z = 8, and r = 4?

24. The natural frequency of a string under constant tension varies inversely as its length. If a string 40 cm long vibrates 680 times per second, what length must the string be to vibrate 850 times per second under the same tension?

25. What is the square of the perimeter of the square whose area is 64 square feet?

26. What is one-third of the perimeter of the square whose area is 36 square units?

27. How many points do the graphs of y = x2 and xy = 27 have in common?

28. The lines with equations x = 2, x = 5, y = 7, and y = 3 form a rectangle. What is the area of this rectangle?

29. The mass of a metal cylinder varies jointly as its height and the square of the radius of its base. One cylinder has a mass of 120 g. Find the mass of a second cylinder made of the same metal, 3 times as high, and having one-half the base radius of the first.

30. A number y varies jointly as x and z. If y = -24 when x = 4 and z = 3, what is y when x = -6 and z = -2?

Part 3
Type the answer to the problem in the text box below each item. Be sure legibly show all work in your notebook. Remember to include any applicable units. If there is no solution, type "no solution". If there is not enough information present to solve the problem, type "not enough information". (Each question is worth two points)
31. The sum of two numbers is 16, and the sum of their squares 146. Find the numbers.

32. The product of two numbers is 1, and the difference of their squares is 3.75. Find the two numbers.

33. Solve the system:

9x2  + 9y2  = 1

x = y + 1


34. Solve the system:

xy = 8

x + y = 6


35. Solve the system:

x2 + y2 = 25

x2  - y2  = 7

Answer
Here's most of them.  I got tired of working them.
1. n^4-2n^2-8 = 0 = (n+2)(n-2)(n^2+2)
2. c^3+c^2-7c-3 + 0 = (c+3)(c^2-2c-1)
3. another root is 1-i
4. x = +/-2, y = +/-2
5. no real solution
6. x = +/-3/(2^1/2), y = +/-(2^1/2)
7. (40,7),(0,-3)
8. (2,2)
9. (-3,2),(-2,3)
12. not possible
13. x = 5^1/2, y = 2(5^1/2)
14. L = 28, W = 20
15. 12
16. x=8, y=3
17. L=24, W=20
18. L=12, W=5
19. L=8, W=6
20. x = -11, y = -7
21. 540
22. -675
24. 32
25. 1024
26. 8
27. 1
28. 12
31. 5 & 11
34. 2 & 4
35. x = +/-4, y = +/-3