Q3 Algebra Review Pre-calculus Name If the sum of the roots of a quadratic equation is b a and the product of the roots is a c. Find a quadratic equation (with integral coefficients) with the following characteristics. 1. Sum of roots: 12 Product of roots: 27 2. Sum of roots: 3 Product of roots: 70 3. Sum of roots: 7/4 Product of roots: 3/8 4. Sum of roots: 29/35 Product of roots: 6/35 5. Sum of roots: 29/10 Product of roots: 21/10 6. Sum of roots: 5 Product of roots: 24 Solve using sign patterning. Write your answer in algebraic form. 7. (5 x)(3 + x) 0 8. 14x 2 38x + 20 < 0 9. 9 16x 2 0 10. 20x 2 19x + 3 0
Solve for each set of variables. 1. x + 3y = 5 x + 4y = 6 4. p 3q = 1 5p + 16q = 5 2. 300x y = 130 200x + y = 120 5. 2x + 3y 4z = 0 3x 6y + 2z = 1 x 6y 2z = 5 3. x + y + z = 4 4x + 5y = 3 y 3z = 10 6. 2a 6b = 5 4b ½c = 6 2.5a + b 2c = 1
Word problems: Identify the variable(s), write equations or inequalities, solve. 1. The lengths of the sides of a right triangle are in the ratio 3:4:5. Find the area of the triangle, if its perimeter is 24 cm. 2. The sides of one square are each 9 cm longer than the sides of another square. Find the dimensions of the two squares if the sum of their perimeters is 100 cm. 3. The sum of the present ages of Kris and his mother is 38 years. Next year his mother will be four times as old as Kris will be. How old are the two now? Person #1 Person #2 Extra info: YEARS AGO PRESENT (NOW) IN THE FUTURE
4. A 7% acid solution is mixed with a 15% acid solution. How much of the weaker solution must be added to 50ml of the stronger solution to produce an 11% acid solution? Solution #1 Solution #2 New Solution AMOUNT x % SOLUTION = AMT OF CHEM 5. How much water (0% acid) must be added to 1 liter of 90% acid solution to make a 70% acid solution? Solution #1 Solution #2 New Solution AMOUNT OF EACH x % SOLUTION = AMOUNT OF CHEMICAL 6. A cocoa merchant blended cocoa worth $7.50 per kilogram with cocoa worth $9.50 per kilogram to make a mixture of 12 kg of cocoa to be priced at $8.00 per kilogram. How many kilograms of each cocoa were used to make the mixture? First item Second item Mixed item AMOUNT x UNIT PRICE = TOTAL
7. At a carnival the number of adult tickets sold was four more than three times the number of child tickets sold. The total sales amounted to $579.0. How many tickets of each kind were sold if adult tickets cost $4.50 and child tickets cost $3.00? Item #1 Item #2 Totals AMOUNT OF EACH x COST = VALUE 8. Mary invested two sums of money, the first at 10% and the second at 12% interest per year. The ratio of the amounts invested was 8:5. Find the amount of each investment if the interest on the first investment was $200 per year more than the interest on the second investment. Item #1 Item #2 Totals AMOUNT OF EACH x COST = VALUE 9. A moneybag contains 120 nickels and dimes, with a total value of $10. How many dimes are in the bag? First coin/bill Second coin/bill Total AMOUNT OF EACH x VALUE OF EACH = TOTAL VALUE
10. Two cars started at the same time from towns that are 615 km apart and met in 5 hours. What was the average speed of each car if one car went 15 km/h faster than the other? First item Second item Extra info: x TIME = DISTANCE 11. During a five-hour endurance race, one cyclist rode 40 km less than twice the distance traveled by another cyclist. If their combined distance was 290 km, find the average speed of each cyclist. First item Second item Extra info: x TIME = DISTANCE 12. Joe drove 20 km at a steady speed to get to the freeway. Once on the freeway, he increased his speed by 40 km/h. and traveled another 75 km. If the entire trip was completed in 1 hour 14 minutes, what was his speed on each part of the trip? First item Second item Extra info: x TIME = DISTANCE
13. Marika traveled 4 km up a river and then back in 1 hour 4 minutes. If she rows 8 km/h in still water (no current), what is the rate of the river s current? First item Second item Extra info: x TIME = DISTANCE Going DOWNSTREAM increases your rate of speed. r+c Going UPSTREAM decreases your rate of speed. r-c 14. Working together, Mrs. Smith and her son can shovel the snow from their sidewalk in 6 minutes. Working by alone, the son would require 16 minutes more than his mother would to shovel the sidewalk. How long does it take Mrs. Smith to shovel the sidewalk working by herself? x TIME = Work Finished First person Second person Extra info: (Job done = 1)
15. Working together, Verna and Sam can paint their living room in 2 hours. Working alone, it would take Verna 3 hours longer than Sam to paint the room. How long would it take Sam to paint the room working by himself? x TIME = Work Finished First person Second person Extra info: (Job done = 1) 16. One pipe can fill a swimming pool in 15 hours; a second pipe requires 18 hours. If the first pipe is opened for 7 hours and then closed when the second pipe is opened, how long will it take the second to finish filling the pool? x TIME = Work Finished First person Second person Extra info: (Job done = 1) 17. Find the least number such that the sum of the number and twice its reciprocal is 121/42.
18. A local commuter train and an express train serve two cities that are 30 km apart. The local train leaves the station daily at 5pm traveling at 90 km/h. Fifteen minutes later the express leaves the same station traveling at 120 km/h. When will the express pass the other train? First item Second item Extra info: x TIME = DISTANCE 19. The perimeter of a rectangle is 18m and the area is 20 meters squared. What are the dimensions of the rectangle? 20. Find three consecutive positive even integers such that the product of the two smaller exceeds the largest by 16.
21. If the second of three consecutive positive integers is added to the product of the first and third, the result is 109. Find the integers. 22. If the first of three consecutive positive integers is added to the product of the second and third, the result is 194. Find the integers. 23. The area of the larger of two equilateral triangles is 25% greater than the area of the smaller. If the length of one side of the larger triangle is 15 cm, find the length of a side of the smaller triangle.
24. The perimeter of a right triangle is 90 cm. The length of the hypotenuse is 41 cm. Find the length of the longer leg. 25. Find the radius and height of a cylinder if the height is 3 less than the radius and the total area is 70π. 26. A beverage machine contains 59 nickels and dimes for a total of $4.55. How many of each coin does the machine contain?
27. A motor boat travels 160 km upstream in 8 hours and it travels the same distance downstream in 5 hours. What is the rate of the boat is still water? 28. The sum of Margaret s and Janet s ages is 24. Four years ago Janet was three times as old as Margaret was then. How old is each person now? 29. Grove City and Chardon are 492 km apart. Miguel left Chardon at 11 am to go to Grove City. At the same time, Preston left Grove City, driving in the direction of Chardon. If Preston was traveling 12 km/h faster than Miguel and they passed each other at 2 pm, how fast was Miguel traveling? 30. Ben and Molly drive in opposite directions. If Ben drives for 5 h and Molly drives for 3 h, they will be 510 km apart. If Moly drives for 5 h and Ben drives for 3 h, they will be 530 km apart. How fast are they driving?