lesson 8 homework practice quadratic functions page 69

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8.E: Quadratic Functions (Exercises)

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• David Arnold
• College of the Redwoods

8.1: Introduction to Radical Notation

1) List all real square roots of \(-400\).

There are no real square roots.

2) List all real square roots of \(64\).

3) List all real square roots of \(-25\).

4) List all real square roots of \(81\).

5) List all real square roots of \(49\).

6) List all real square roots of \(-100\).

7) List all real square roots of \(324\).

8) List all real square roots of \(36\).

9) List all real square roots of \(-225\).

10) List all real square roots of \(0\).

11) List all real solutions of \(x^2 = -225\).

There are no real solutions.

12) List all real solutions of \(x^2 = -25\).

13) List all real solutions of \(x^2 = 361\).

14) List all real solutions of \(x^2 = 256\).

15) List all real solutions of \(x^2 = -400\).

16) List all real solutions of \(x^2 = 0\).

17) List all real solutions of \(x^2 = 169\).

18) List all real solutions of \(x^{2}=-100\).

19) List all real solutions of \(x^{2}=625\).

20) List all real solutions of \(x^{2}=324\).

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

21) \(\sqrt{64}\)

22) \(-\sqrt{-529}\)

23) \(-\sqrt{-256}\)

The expression is not a real number.

24) \(\sqrt{-529}\)

25) \(-\sqrt{361}\)

26) \(\sqrt{-361}\)

27) \(-\sqrt{100}\)

28) \(-\sqrt{196}\)

29) \(\sqrt{441}\)

30) \(\sqrt{49}\)

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

31) \((-\sqrt{17})^{2}\)

32) \((-\sqrt{31})^{2}\)

33) \((\sqrt{59})^{2}\)

34) \((\sqrt{43})^{2}\)

35) \((-\sqrt{29})^{2}\)

36) \((-\sqrt{89})^{2}\)

37) \((\sqrt{79})^{2}\)

38) \((\sqrt{3})^{2}\)

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) \(x^{2}=37\)

\(\pm \sqrt{37} \approx \pm 6.082763\)

40) \(x^{2}=32\)

41) \(x^{2}=11\)

\(\pm \sqrt{11} \approx \pm 3.316625\)

42) \(x^{2}=42\)

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) \(\sqrt{5} \sqrt{13}\)

\(\sqrt{65}\)

2) \(\sqrt{2} \sqrt{7}\)

3) \(\sqrt{17} \sqrt{2}\)

\(\sqrt{34}\)

4) \(\sqrt{5} \sqrt{11}\)

5) \(\sqrt{5} \sqrt{17}\)

\(\sqrt{85}\)

6) \(\sqrt{17} \sqrt{3}\)

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

7) \(\sqrt{56}\)

\(2 \sqrt{14}\)

8) \(\sqrt{45}\)

9) \(\sqrt{99}\)

\(3 \sqrt{11}\)

10) \(\sqrt{75}\)

11) \(\sqrt{150}\)

\(5 \sqrt{6}\)

12) \(\sqrt{90}\)

13) \(\sqrt{40}\)

\(2 \sqrt{10}\)

14) \(\sqrt{171}\)

15) \(\sqrt{28}\)

\(2 \sqrt{7}\)

16) \(\sqrt{175}\)

17) \(\sqrt{153}\)

\(3 \sqrt{17}\)

18) \(\sqrt{125}\)

19) \(\sqrt{50}\)

\(5 \sqrt{2}\)

20) \(\sqrt{88}\)

21) \(\sqrt{18}\)

\(3 \sqrt{2}\)

22) \(\sqrt{117}\)

23) \(\sqrt{44}\)

\(2 \sqrt{11}\)

24) \(\sqrt{20}\)

25) \(\sqrt{104}\)

\(2 \sqrt{26}\)

26) \(\sqrt{27}\)

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

\(2 \sqrt{15}\)

\(2 \sqrt{154}\)

\(2 \sqrt{37}\)

\(2 \sqrt{74}\)

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

\(\dfrac{25}{8} \pi-6\)

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

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.

\(7,24,25\)

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.

\(18.3\) 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) \(x^{2}=84\)

\(\pm 2 \sqrt{21}\)

2) \(x^{2}=88\)

3) \(x^{2}=68\)

\(\pm 2 \sqrt{17}\)

4) \(x^{2}=112\)

5) \(x^{2}=-16\)

No real solutions

6) \(x^{2}=-104\)

7) \(x^{2}=124\)

\(\pm 2 \sqrt{31}\)

8) \(x^{2}=148\)

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

9) \((x+19)^{2}=36\)

\(-25,-13\)

10) \((x-4)^{2}=400\)

11) \((x+14)^{2}=100\)

12) \((x-15)^{2}=100\)

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

13) \((x+23)^{2}\)

\(x^{2}+46 x+529\)

14) \((x-5)^{2}\)

15) \((x+11)^{2}\)

\(x^{2}+22 x+121\)

16) \((x-7)^{2}\)

17) \((x-25)^{2}\)

\(x^{2}-50 x+625\)

18) \((x+4)^{2}\)

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

19) \(x^{2}+24 x+144\)

\((x+12)^{2}\)

20) \(x^{2}-16 x+64\)

21) \(x^{2}-34 x+289\)

\((x-17)^{2}\)

22) \(x^{2}+8 x+16\)

23) \(x^{2}-20 x+100\)

\((x-10)^{2}\)

24) \(x^{2}+16 x+64\)

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) \(x^{2}-20 x\)

\(x^{2}-20 x+100\)

26) \(x^{2}-10 x\)

27) \(x^{2}-6 x\)

\(x^{2}-6 x+9\)

28) \(x^{2}-40 x\)

29) \(x^{2}+20 x\)

\(x^{2}+20 x+100\)

30) \(x^{2}+26 x\)

31) \(x^{2}+7 x\)

\(x^{2}+7 x+\frac {49}{4}\)

32) \(x^{2}+19 x\)

33) \(x^{2}+15 x\)

\(x^{2}+15 x+\frac {225}{4}\)

34) \(x^{2}+25 x\)

35) \(x^{2}-5 x\)

\(x^{2}-5 x+\frac {25}{4}\)

36) \(x^{2}-3 x\)

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

37) \(x^{2}=18 x-18\)

\(9-3 \sqrt{7}, 9+3 \sqrt{7}\)

38) \(x^{2}=12 x-18\)

39) \(x^{2}=16 x-16\)

\(8-4 \sqrt{3}, 8+4 \sqrt{3}\)

40) \(x^{2}=12 x-4\)

41) \(x^{2}=-16 x-4\)

\(-8-2 \sqrt{15},-8+2 \sqrt{15}\)

42) \(x^{2}=-12 x-12\)

43) \(x^{2}=18 x-9\)

\(9-6 \sqrt{2}, 9+6 \sqrt{2}\)

44) \(x^{2}=16 x-10\)

45) \(x^{2}=16 x-8\)

\(8-2 \sqrt{14}, 8+2 \sqrt{14}\)

46) \(x^{2}=10 x-5\)

47) \(x^{2}=-18 x-18\)

\(-9-3 \sqrt{7},-9+3 \sqrt{7}\)

48) \(x^{2}=-10 x-17\)

49) \(x^{2}=-16 x-20\)

\(-8-2 \sqrt{11},-8+2 \sqrt{11}\)

50) \(x^{2}=-16 x-12\)

51) \(x^{2}=-18 x-1\)

\(-9-4 \sqrt{5},-9+4 \sqrt{5}\)

52) \(x^{2}=-12 x-8\)

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) \(x^{2}-2 x-17=0\)

\(1-3 \sqrt{2}, 1+3 \sqrt{2}\)

54) \(x^{2}-4 x-14=0\)

55) \(x^{2}-6 x-3=0\)

\(3-2 \sqrt{3}, 3+2 \sqrt{3}\)

56) \(x^{2}-4 x-16=0\)

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) \(x^{2}-3 x-28=0\)

2) \(x^{2}-4 x-12=0\)

3) \(x^{2}-8 x+15=0\)

4) \(x^{2}-6 x+8=0\)

5) \(x^{2}-2 x-48=0\)

6) \(x^{2}+9 x+8=0\)

7) \(x^{2}+x-30=0\)

8) \(x^{2}-17 x+72=0\)

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) \(x^{2}-7 x-5=0\)

\(\dfrac{7 \pm \sqrt{69}}{2}\)

10) \(3 x^{2}-3 x-4=0\)

11) \(2 x^{2}+x-4=0\)

\(\dfrac{-1 \pm \sqrt{33}}{4}\)

12) \(2 x^{2}+7 x-3=0\)

13) \(x^{2}-7 x-4=0\)

\(\dfrac{7 \pm \sqrt{65}}{2}\)

14) \(x^{2}-5 x+1=0\)

15) \(4 x^{2}-x-2=0\)

\(\dfrac{1 \pm \sqrt{33}}{8}\)

16) \(5 x^{2}+x-2=0\)

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) \(x^{2}-x-11=0\)

\(\dfrac{1 \pm 3 \sqrt{5}}{2}\)

18) \(x^{2}-11 x+19=0\)

19) \(x^{2}-9 x+9=0\)

\(\dfrac{9 \pm 3 \sqrt{5}}{2}\)

20) \(x^{2}+5 x-5=0\)

21) \(x^{2}-3 x-9=0\)

\(\dfrac{3 \pm 3 \sqrt{5}}{2}\)

22) \(x^{2}-5 x-5=0\)

23) \(x^{2}-7 x-19=0\)

\(\dfrac{7 \pm 5 \sqrt{5}}{2}\)

24) \(x^{2}+13 x+4=0\)

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) \(12 x^{2}+10 x-1=0\)

\(\dfrac{-5 \pm \sqrt{37}}{12}\)

26) \(7 x^{2}+6 x-3=0\)

27) \(7 x^{2}-10 x+1=0\)

\(\dfrac{5 \pm 3 \sqrt{2}}{7}\)

28) \(7 x^{2}+4 x-1=0\)

29) \(2 x^{2}-12 x+3=0\)

\(\dfrac{6 \pm \sqrt{30}}{2}\)

30) \(2 x^{2}-6 x-13=0\)

31) \(13 x^{2}-2 x-2=0\)

\(\dfrac{1 \pm 3 \sqrt{3}}{13}\)

32) \(9 x^{2}-2 x-3=0\)

33) An object is launched vertically and its height \(y\) (in feet) above ground level is given by the equation \(y = 240 + 160t− 16t^2\), 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.

\(11.3\) seconds

34) An object is launched vertically and its height \(y\) (in feet) above ground level is given by the equation \(y = 192 + 288t− 16t^2\), 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 \(R = 6000x − 5x^2\). 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.

\(90\) widgets, \(1109\) widgets

36) A manufacturer’s revenue \(R\) accrued from selling \(x\) widgets is given by the equation \(R = 4500x−15.25x^2\). 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.

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.

\(5.9\) 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.

A quadratic model is fitted to this data, yielding \[C =0 .01125t^2 +0 .925t+ 318 \nonumber \] 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.

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

A quadratic model is fitted to this data, yielding \[P =0 .01375t^2 +0 .0525t+1 .4 \nonumber \] 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.

Curriculum  /  Math  /  9th Grade  /  Unit 7: Quadratic Functions and Solutions  /  Lesson 8

Quadratic Functions and Solutions

Lesson 8 of 13

Criteria for Success

Tips for teachers, anchor problems, problem set, target task, additional practice.

Factor special cases of quadratic equations—difference of two squares.

Common Core Standards

Core standards.

The core standards covered in this lesson

Seeing Structure in Expressions

A.SSE.A.1.A — Interpret parts of an expression, such as terms, factors, and coefficients.

A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ — y⁴ as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).

A.SSE.B.3.A — Factor a quadratic expression to reveal the zeros of the function it defines.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Identify features of two linear binomials that when multiplied together result in a quadratic binomial difference of two squares. 
• Factor and solve quadratic equations that represent a difference of two squares.
• Describe graphical features of quadratic functions that are differences of two squares.

Suggestions for teachers to help them teach this lesson

Lessons 8 and 9 look at specific cases of factoring quadratic expressions—difference of two squares and perfect square trinomials. Depending on the needs of your students, these lessons can be kept separate or combined together. These lessons are also a good opportunity to spiral in other factoring problems from Lessons 4–7. 

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Aaron says that when you multiply two linear binomials, you will always get a trinomial. Alison disagrees and finds an example where two linear binomials multiplied together produce a quadratic binomial. 

Find an example that demonstrates Alison’s claim is true.

Guiding Questions

Find the solutions to the quadratic equations below. 

a.   $${y=x^2-196}$$

b.   $${y=16x^2-121}$$

c.   $${y=75x^2-27 }$$

Graphs of two quadratic functions are shown below. 

Which graph represents a quadratic function whose equation is a difference of two squares? Explain how you know.

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Factor each expression completely.

a.   $${x^2-36}$$

b.   $${12x^2-3}$$

c.   $${6x^2+34x-12}$$

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• Include spiraled problems that cover various factoring examples seen so far
• EngageNY Mathematics Algebra I > Module 4 > Topic A > Lesson 1 — Exercises 4-7
• EngageNY Mathematics Algebra I > Module 4 > Topic A > Lesson 3 — Problem Set
• Mathematics Vision Project: Secondary Mathematics Two Module 3: Quadratic Equations — Lesson 3.4 "Go"

Topic A: Features of Quadratic Functions

Compare quadratic, exponential, and linear functions represented as graphs, tables, and equations.

F.IF.B.4 F.LE.A.2

Identify key features of a quadratic function represented graphically. Graph a quadratic function from a table of values.

F.IF.B.4 F.IF.C.7.A

Calculate and compare the average rate of change for linear, exponential, and quadratic functions.

F.IF.B.4 F.IF.B.6 F.LE.A.3

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Topic B: Factoring and Solutions of Quadratic Equations

Factor quadratic expressions using the greatest common factor. Demonstrate equivalence between expressions by multiplying polynomials.

A.APR.A.1 A.SSE.A.2 A.SSE.B.3.A

 Identify solutions to quadratic equations using the zero product property (equations written in intercept form).

A.APR.B.3 F.IF.C.8.A

Factor quadratic equations and identify solutions (when leading coefficient is equal to 1).

A.SSE.A.1.A A.SSE.B.3.A

Factor quadratic equations and identify solutions (when leading coefficient does not equal 1).

A.SSE.A.1.A A.SSE.A.2 A.SSE.B.3.A

Factor special cases of quadratic equations—perfect square trinomials.

A.SSE.A.2 A.SSE.B.3.A

Solve quadratic equations by factoring. Compare solutions in different representations (graph, equation, and table).

A.SSE.B.3.A F.IF.C.8.A F.IF.C.9

Solve quadratic equations by taking square roots. 

A.REI.B.4.B

Graph quadratic functions using $${x-}$$ intercepts and vertex.

A.APR.B.3 F.IF.B.4 F.IF.C.7.A F.IF.C.8.A

Topic C: Interpreting Solutions of Quadratic Functions in Context

Interpret quadratic solutions in context.

A.CED.A.1 F.IF.B.4 F.IF.B.5 F.IF.C.8.A

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Chapter 8, Lesson 6: Quadratic Equations: Perfect Squares

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