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Mathematics LibreTexts

8.E: Quadratic Functions (Exercises)

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8.1: Introduction to Radical Notation

1) List all real square roots of 400.

Answer

There are no real square roots.

2) List all real square roots of 64.

3) List all real square roots of 25.

Answer

There are no real square roots.

4) List all real square roots of 81.

5) List all real square roots of 49.

Answer

7,7

6) List all real square roots of 100.

7) List all real square roots of 324.

Answer

18,18

8) List all real square roots of 36.

9) List all real square roots of 225.

Answer

There are no real square roots.

10) List all real square roots of 0.

11) List all real solutions of x2=225.

Answer

There are no real solutions.

12) List all real solutions of x2=25.

13) List all real solutions of x2=361.

Answer

19,19

14) List all real solutions of x2=256.

15) List all real solutions of x2=400.

Answer

There are no real solutions.

16) List all real solutions of x2=0.

17) List all real solutions of x2=169.

Answer

13,13

18) List all real solutions of x2=100.

19) List all real solutions of x2=625.

Answer

25,25

20) List all real solutions of x2=324.

In Exercises 21-30, simplify each of the given expressions.

21) 64

Answer

8

22) 529

23) 256

Answer

The expression is not a real number.

24) 529

25) 361

Answer

19

26) 361

27) 100

Answer

10

28) 196

29) 441

Answer

21

30) 49

In Exercises 31-38, simplify each of the given expressions.

31) (17)2

Answer

17

32) (31)2

33) (59)2

Answer

59

34) (43)2

35) (29)2

Answer

29

36) (89)2

37) (79)2

Answer

79

38) (3)2

In Exercises 39-42, for each of the given equations, first use the 5:intersect utility on the CALC menu of the graphing calculator to determine the solutions. Follow the Calculator Submission Guidelines, as demonstrated in Example 8.1.9 in reporting the solution on your homework paper. Second, solve the equation algebraically, then use your calculator to find approximations of your answers and compare this second set with the first set of answers.

39) x2=37

Answer

Answer 8.1.39.png

±37±6.082763

40) x2=32

41) x2=11

Answer

Answer 8.1.41.png

±11±3.316625

42) x2=42

8.2: Simplifying Radical Expressions

In Exercises 1-6, simplify the given expression, writing your answer using a single square root symbol. Check the result with your graphing calculator.

1) 513

Answer

65

2) 27

3) 172

Answer

34

4) 511

5) 517

Answer

85

6) 173

In Exercises 7-26, convert each of the given expressions to simple radical form.

7) 56

Answer

214

8) 45

9) 99

Answer

311

10) 75

11) 150

Answer

56

12) 90

13) 40

Answer

210

14) 171

15) 28

Answer

27

16) 175

17) 153

Answer

317

18) 125

19) 50

Answer

52

20) 88

21) 18

Answer

32

22) 117

23) 44

Answer

211

24) 20

25) 104

Answer

226

26) 27

In Exercises 27-34, find the length of the missing side of the right triangle. Your final answer must be in simple radical form.

27)

Exercise 8.2.27.png
Answer

215

28)

Exercise 8.2.28.png

29)

Exercise 8.2.29.png
Answer

2154

30)

Exercise 8.2.30.png

31)

Exercise 8.2.31.png
Answer

237

32)

Exercise 8.2.32.png

33)

Exercise 8.2.33.png
Answer

274

34)

Exercise 8.2.34.png

35) In the figure below, a right triangle is inscribed in a semicircle. What is the area of the shaded region?

Exercise 8.2.35.png
Answer

258π6

36) In the figure below, a right triangle is inscribed in a semicircle. What is the area of the shaded region?

Exercise 8.2.36.png

37) The longest leg of a right triangle is 10 feet longer than twice the length of its shorter leg. The hypotenuse is 4 feet longer than three times the length of the shorter leg. Find the lengths of all three sides of the right triangle.

Answer

7,24,25

38) The longest leg of a right triangle is 2 feet longer than twice the length of its shorter leg. The hypotenuse is 3 feet longer than twice the length of the shorter leg. Find the lengths of all three sides of the right triangle.

39) A ladder 19 feet long leans against the garage wall. If the base of the ladder is 5 feet from the garage wall, how high up the garage wall does the ladder reach? Use your calculator to round your answer to the nearest tenth of a foot.

Answer

18.3 feet

40) A ladder 19 feet long leans against the garage wall. If the base of the ladder is 6 feet from the garage wall, how high up the garage wall does the ladder reach? Use your calculator to round your answer to the nearest tenth of a foot.

8.3: Completing the Square

In Exercises 1-8, find all real solutions of the given equation. Place your final answers in simple radical form.

1) x2=84

Answer

±221

2) x2=88

3) x2=68

Answer

±217

4) x2=112

5) x2=16

Answer

No real solutions

6) x2=104

7) x2=124

Answer

±231

8) x2=148

In Exercises 9-12, find all real solutions of the given equation. Place your final answers in simple radical form.

9) (x+19)2=36

Answer

25,13

10) (x4)2=400

11) (x+14)2=100

Answer

24,4

12) (x15)2=100

In Exercises 13-18, square each of the following binomials.

13) (x+23)2

Answer

x2+46x+529

14) (x5)2

15) (x+11)2

Answer

x2+22x+121

16) (x7)2

17) (x25)2

Answer

x250x+625

18) (x+4)2

In Exercises 19-24, factor each of the following trinomials.

19) x2+24x+144

Answer:

(x+12)2

20) x216x+64

21) x234x+289

Answer:

(x17)2

22) x2+8x+16

23) x220x+100

Answer:

(x10)2

24) x2+16x+64

In Exercises 25-36, for each expression, complete the square to form a perfect square trinomial. Check your answer by factoring your result. Be sure to check your middle term.

25) x220x

Answer

x220x+100

26) x210x

27) x26x

Answer

x26x+9

28) x240x

29) x2+20x

Answer

x2+20x+100

30) x2+26x

31) x2+7x

Answer

x2+7x+494

32) x2+19x

33) x2+15x

Answer

x2+15x+2254

34) x2+25x

35) x25x

Answer

x25x+254

36) x23x

In Exercises 37-52, find all real solutions, if any, of the given equation. Place your final answers in simple radical form.

37) x2=18x18

Answer

937,9+37

38) x2=12x18

39) x2=16x16

Answer

843,8+43

40) x2=12x4

41) x2=16x4

Answer

8215,8+215

42) x2=12x12

43) x2=18x9

Answer

962,9+62

44) x2=16x10

45) x2=16x8

Answer

8214,8+214

46) x2=10x5

47) x2=18x18

Answer

937,9+37

48) x2=10x17

49) x2=16x20

Answer

8211,8+211

50) x2=16x12

51) x2=18x1

Answer

945,9+45

52) x2=12x8

In Exercises 53-56, solve the given equation algebraically, stating your final answers in simple radical form. Next, use the graphing calculator to solve the equation, following the technique outlined in Example 8.3.8. Use the Calculator Submission Guidelines, as demonstrated inExample 8, when reporting the solution on your homework. Compare the solutions determined by the two methods.

53) x22x17=0

Answer

132,1+32

54) x24x14=0

55) x26x3=0

Answer

323,3+23

56) x24x16=0

8.4: The Quadratic Formula

In Exercises 1-8, solve the given equation by factoring the trinomial using the ac-method, then applying the zero product property. Secondly, craft a second solution using the quadratic formula. Compare your answers.

1) x23x28=0

Answer

4,7

2) x24x12=0

3) x28x+15=0

Answer

3,5

4) x26x+8=0

5) x22x48=0

Answer

6,8

6) x2+9x+8=0

7) x2+x30=0

Answer

6,5

8) x217x+72=0

In Exercises 9-16, use the quadratic formula to solve the given equation. Your final answers must be reduced to lowest terms and all radical expressions must be in simple radical form.

9) x27x5=0

Answer

7±692

10) 3x23x4=0

11) 2x2+x4=0

Answer

1±334

12) 2x2+7x3=0

13) x27x4=0

Answer

7±652

14) x25x+1=0

15) 4x2x2=0

Answer

1±338

16) 5x2+x2=0

In Exercises 17-24, use the quadratic formula to solve the given equation. Your final answers must be reduced to lowest terms and all radical expressions must be in simple radical form.

17) x2x11=0

Answer

1±352

18) x211x+19=0

19) x29x+9=0

Answer

9±352

20) x2+5x5=0

21) x23x9=0

Answer

3±352

22) x25x5=0

23) x27x19=0

Answer

7±552

24) x2+13x+4=0

In Exercises 25-32, use the quadratic formula to solve the given equation. Your final answers must be reduced to lowest terms and all radical expressions must be in simple radical form.

25) 12x2+10x1=0

Answer

5±3712

26) 7x2+6x3=0

27) 7x210x+1=0

Answer

5±327

28) 7x2+4x1=0

29) 2x212x+3=0

Answer

6±302

30) 2x26x13=0

31) 13x22x2=0

Answer

1±3313

32) 9x22x3=0

33) An object is launched vertically and its height y (in feet) above ground level is given by the equation y=240+160t16t2, where t is the time (in seconds) that has passed since its launch. How much time must pass after the launch before the object returns to ground level? After placing the answer in simple form and reducing, use your calculator to round the answer to the nearest tenth of a second.

Answer

11.3 seconds

34) An object is launched vertically and its height y (in feet) above ground level is given by the equation y=192+288t16t2, where t is the time (in seconds) that has passed since its launch. How much time must pass after the launch before the object returns to ground level? After placing the answer in simple form and reducing, use your calculator to round the answer to the nearest tenth of a second.

35) A manufacturer’s revenue R accrued from selling x widgets is given by the equation R=6000x5x2. The manufacturer’s costs for building x widgets is given by the equation C=500000+5.25x. The break even point for the manufacturer is defined as the number of widgets built and sold so the the manufacturer’s revenue and costs are identical. Find the number of widgets required to be built and sold so that the manufacturer “breaks even.” Round your answers to the nearest widget.

Answer

90 widgets, 1109 widgets

36) A manufacturer’s revenue R accrued from selling x widgets is given by the equation R=4500x15.25x2. How many widgets must be sold so that the manufacturer’s revenue is $125,000? Round your answers to the nearest widget.

37) Mike gets on his bike at noon and begins to ride due north at a constant rate of 6 miles per hour. At 2:00 pm, Todd gets on his bike at the same starting point and begins to ride due east at a constant rate of 8 miles per hour. At what time of the day will they be 60 miles apart (as the crow flies)? Don’t worry about simple form, just report the time of day, correct to the nearest minute.

Answer

7:12 pm

38) Mikaela gets on her bike at noon and begins to ride due north at a constant rate of 4 miles per hour. At 1:00 pm, Rosemarie gets on her bike at the same starting point and begins to ride due east at a constant rate of 6 miles per hour. At what time of the day will they be 20 miles apart (as the crow flies)? Don’t worry about simple form, just report the time of day, correct to the nearest minute.

39) The area of a rectangular field is 76 square feet. The length of the field is 7 feet longer than its width. Find the dimensions of the field, correct to the nearest tenth of a foot.

Answer

5.9 by 12.9 feet

40) The area of a rectangular field is 50 square feet. The length of the field is 8 feet longer than its width. Find the dimensions of the field, correct to the nearest tenth of a foot.

41) Mean concentrations of carbon dioxide over Mauna Loa, Hawaii, are gathered by the Earth System Research Laboratory (ESRL) in conjunction with the National Oceanic and Atmosphere Administration (NOAA). Mean annual concentrations in parts per million for the years 1962, 1982, and 2002 are shown in the following table.

Year 1962 1982 2002
Concentration (ppm) 318 341 373

A quadratic model is fitted to this data, yielding C=0.01125t2+0.925t+318 where t is the number of years since 1962 and C is the mean annual concentration (in parts per million) of carbon dioxide over Mauna Loa. Use the model to find the year when the mean concentration of carbon dioxide was 330 parts per million. Round your answer to the nearest year.

Answer

1973

42) The U.S. Census Bureau provides historical data on the number of Americans over the age of 85.

Year 1970 1990 2010
Population over 85 (millions) 1.4 3.0 5.7

A quadratic model is fitted to this data, yielding P=0.01375t2+0.0525t+1.4 where t is the number of years since 1970 and P is number of Americans (in millions) over the age of 85. Use the model to find the year when the number of Americans over the age of 85 was 2,200,000. Round your answer to the nearest year.


This page titled 8.E: Quadratic Functions (Exercises) is shared under a CC BY-NC-ND 3.0 license and was authored, remixed, and/or curated by David Arnold via source content that was edited to the style and standards of the LibreTexts platform.

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