the quadratic formula 7 - high school math | prek...

LESSON OBJECTIVES

� Create a histogram and a stem-and-leaf plot of a data set� Given a list of data, use a calculator to graph a histogram� Interpret histograms and stem-and-leaf plots� Decide the appropriateness of a histogram and a stem-and-

leaf plot for a given data set

OBJECTIVES

� Use the vertex form of a quadratic equation to find the equation’s roots

� Derive the quadratic formula by completing the square

� Use the quadratic formula to solve application problems

OUTLINE

One day: 15 min Example A

10 min Investigation

10 min Discuss Investigation

5 min Example B

5 min Exercises

MATERIALS

� Investigation Worksheet, optional

� Calculator Note 7D, optional

� Sketchpad demonstration Quadratic Functions, optional

ADDITIONAL SUPPORT

� Lesson 7.4 More Practice Your Skills

� Lesson 7.4 Condensed Lessons (in English or Spanish)

� TestCheck worksheets

TEACHINGTHE LESSON

This lesson derives the quadratic formula. Students who used Discovering Algebra may recall it from Chapter 9 of that book.

ONGOING ASSESSMENT

Check students’ familiarity with the general form of a quadratic and with the projectile motion function. Also see how well they remember the quadratic formula from previous courses.

DIFFERENTIATING INSTRUCTION

ELL

Students should be able to differentiate among quadratic functions (the relation ship between two variables), quadratic equations (a specific value of the func tions), and the quadratic formula (a method used to solve quadratic equations). Have students provide verbal and symbolic examples of each.

Extra Support

Students should work through Example A as well as another example or two like it before looking at the quadratic for-mula’s derivation. Emphasize the links among the derivation, the formula, and solutions.

Advanced

Ask students to derive the qua-dratic formula on their own after they see Example A. Have students explain their insights into the relationship between the constant values of a, b, and c and the solutions for x.

The Quadratic FormulaAlthough you can always use a graph of a quadratic function to approximate the x-intercepts, you are often not able to find exact solutions. This lesson will develop a procedure to find the exact roots of a quadratic equation. Consider again this situation from Example C in the previous lesson.

Nora hits a softball straight up at a speed of 120 ft/s. Her bat contacts the ball at a height of 3 ft above the ground. Recall that the equation relating height in feet, y, and time in seconds, x, is y � �16x2 � 120x � 3. How long will it be until the ball hits the ground?

The height will be zero when the ball hits the ground, so you want to find the solutions to the equation �16x2 � 120x � 3 � 0. You can approximate the x-intercepts by graphing, but you may not be able to find the exact x-intercept.

This value represents �5.59908 � 10�10,or �0.000000000559908..., which isvery close to zero.

You will not be able to factor this equation using a rectangle diagram, so you can’t use the zero-product property. Instead, to solve this equation symbolically, first write the equation in the form a(x � h)2 � k � 0.

�16x2 � 120x � 3 � 0 Original equation.

�16x2 � 120x � �3 Subtract the constant from

both sides.

�16�x2 � 7.5x � ? � � � �3 Factor to get the leading

coefficient 1.

�16 �x2 � 7.5x � ��3.75 � 2� � �3 � �16��3.75 � 2 Complete the square.

�16�x � 3.75� 2 � �228 Factor and combine like terms.

�x � 3.75� 2 � 14.25 Divide by �16.

x � 3.75 � � � _____

14.25 Take the square root of both sides.

x � 3.75 � � _____

14.25 Add 3.75 to both sides.

x � 3.75 � � _____

14.25 or x � 3.75 � � _____

14.25 Write the two exact solutions to the equation.

x 7.525 or x �0.025 Approximate the values of x.

The zeros of the function are x 7.525 and x �0.025. The negative time, �0.025 s, does not make sense in this situation, so the ball hits the ground after approximately 7.525 s.

EXAMPLE AEXAMPLE A

� Solution� Solution

L E S S O N

7.4

L E S S O N

7.4

LESSON 7.4 The Quadratic Formula 403

DAA2TE_985_07a.indd 403DAA2TE_985_07a.indd 403 12/30/08 2:31:25 PM12/30/08 2:31:25 PM

404 CHAPTER 7 Quadratic and Other Polynomial Functions

Discussing the Lesson

LESSON EXAMPLE A

This example revisits the situation of Lesson 7.3, Exam ple C, but it poses a problem that reviews com-pleting the square and extends it to solving an equation. Resist the temptation to skip this example and go right to the quadratic formula; the example provides a special case for the derivation of that formula.

In applying the quadratic for-mula to Example A, you might want to simplify to the exact

solutions, x � 3.75 � � ___

57 ____ 2 .

If students won’t be doing the project at the end of this lesson, you might let them use the cal-culator program found in Calculator Note 7D.

[Alert] Most mistakes made while using a calculator with the quadratic formula can be blamed on not enclosing the radicand within parentheses, not enclosing the entire numerator within parentheses, or not enclosing the denominator within parentheses.

Guiding the Investigation

This is a deepening skills investigation. The investigation prompts students to examine the validity of their solutions in context. Students should develop this habit of figuring out not just “what” but also “if” and “why.”

MODIFYING THE INVESTIGATION

Whole Class Complete Steps 1 through 6 with student input.

Shortened Skip Step 6.

One Step Remind students of the second part of the One Step investigation of Lesson 7.3, in which they knew the initial velocity of a baseball. Ask them when the ball hits the ground. Students can use the vertex form they found, set it equal to 0, and solve for time. Or they can double the x-coordinate of

the vertex. Challenge them to solve the general equation ax2 � bx � c � 0 by completing the square and to come up with a formula for the solution to any quadratic equation. Don’t be satisfied if they already can reel off the quadratic formula; the goal is for them to derive it.

FACILITATING STUDENT WORK

Step 1 Students are to use the projectile motion function from Lesson 7.3.

Step 4 A graph of the function can help students answer this question.

Step 5 In Step 3, the symmetry of the two solutions about the x-coordinate of the vertex is consistent with the symmetry of the parabola.

If you follow the same steps with a general quadratic equation, then you can develop the quadratic formula. This formula provides solutions to ax2 � bx � c � 0 in terms of a, b, and c.

ax2 � bx � c � 0 Original equation.

ax2 � bx � �c Subtract c from both sides.

a � x 2 � b __ a x � ? � � � �c Factor to get the leading coefficient 1.

a � x � b __ a x � � b ___ 2a � 2 � � a � b ___ 2a �

2 � c Complete the square.

a � x � b ___ 2a � 2 � b 2 ___ 4a � c Factor the perfect-square trinomial on the

left side and multiply on the right side.

a � x � b ___ 2a � 2 � b 2 ___ 4a � 4ac ___ 4a Rewrite the right side with a common

denominator.

a � x � b ___ 2a � 2 � b 2 � 4ac _______ 4a Add terms with a common denominator.

� x � b ___ 2a � 2 � b 2 � 4ac _______

4a 2 Divide both sides by a.

x � b ___ 2a � � � ________

b 2 � 4ac _______ 4a 2

Take the square root of both sides.

x � b ___ 2a � � � ________

b 2 � 4ac __________ 2a Use the power of a quotient property to

take the square roots of the numerator

and denominator.

x � � b ___ 2a � � ________

b 2 � 4ac __________ 2a Subtract b __ 2a from both sides.

x � �b � � ________

b 2 � 4ac ______________ 2a Add terms with a common denominator.

The Quadratic Formula

Given a quadratic equation written in the form ax2 � bx � c � 0, the solutions are

x � �b � � ________

b 2 � 4ac ______________ 2a

To use the quadratic formula on the equation in Example A, �16x2 � 120x � 3 � 0, first identify the coefficients as a � �16, b � 120, and c � 3. The solutions are

x � �120 �

� _______________

120 2 � 4(�16)(3) ________________________

2(�16)

x � �120 � � _____

14592 ______________ �32 or x � �120 � �

_____ 14592 ______________

�32

x �0.025 or x 7.525

The quadratic formula gives you a way to find the roots of any equation in the form ax2 � bx � c � 0. The investigation will give you an opportunity to apply the quadratic formula in different situations.


Step 6 This step foreshadows the introduction of complex numbers in Lesson 7.5. [Alert] Watch for students who fudge their numbers to get real number solutions.

ASSESSING PROGRESS

Give special attention to how students are sub-stituting a, b, and c into the quadratic formula and simplifying.

DISCUSSING THE INVESTIGATION

[Language] During student presentations, discour-age the use of the term quadratic equation to mean quadratic formula. The quadratic formula can be used to solve all quadratic equations.

[Critical Question] “What use is the quadratic form-ula?” [Big Idea] The quadratic formula provides a shortcut for finding the zeros of any quadratic function written in general form, without having to complete the square.

[Ask] “Why are the two solutions in the form of a certain amount added to and subtracted from another number?” [The “other number” is �b

___ 2a , the x-coordinate of the parabola’s vertex. The sym-metry of the parabola ensures that its x-intercepts will be equidistant on either side of this coordinate.]

Ask whether the value of k, the y-value of the vertex, also appears in the quadratic formula. �The number being added to and sub-tracted from �b

___ 2a is the square root of � k _ a .�Remind students that they can gain insight from looking at extreme cases. [Ask] “When the value of a, b, or c is 0, what can be said about the equation, and what happens to the formula?” [When a � 0, the equation is lin-ear, not quadratic, because the quadratic formula requires divid-ing by a, the result when a � 0 is not defined; the formula doesn’t apply to linear equations. When b � 0, the equation’s graph is symmetric about the y-axis; the formula gives two solutions equidistant from 0 or a double solution at x � 0. When c � 0, the equation factors to show a root at x � 0, and the formula returns 0 as one of the roots.]

[Ask] “What if the number repre-sented by b2 � 4ac in the square root is 0?” �Both solutions are the

same, �b ___ 2a . The parabolic graph of

the quadratic function touches the x-axis at just one point, a point of

tangency at the vertex: ��2a

b�, 0�.�

You might describe the quantity b2 � 4ac as the discriminant.

[Ask] “What if the number repre-sented by b2 � 4ac is negative?” [The equation has no real solu-tions, indicating that its parabolic graph doesn’t cross the x-axis.]

You might use the Sketchpad demonstration Quadratic Functions to investigate how various values of a, b, and c affect the graph of a quadratic function.

Salvador hits a baseball at a height of 3 ft and with an initial upward velocity of 88 feet per second.

Step 1 Let x represent time in seconds after the ball is hit, and let y represent the height of the ball in feet. Write an equation that gives the height as a function of time.

Step 2 Write an equation to find the times when the ball is 24 ft above the ground.

Step 3 Rewrite your equation from Step 2 in the form ax2 � bx � c � 0, then use the quadratic formula to solve. What is the real-world meaning of each of your solutions? Why are there two solutions?

Step 4 Find the y-coordinate of the vertex of this parabola. How many different x-values correspond to this y-value? Explain.

Step 5 Write an equation to find the time when the ball reaches its maximum height. Use the quadratic formula to solve the equation. At what point in the solution process does it become obvious that there is only one solution to this equation?

Step 6 Write an equation to find the time when the ball reaches a height of 200 ft. What happens when you try to solve this impossible situation with the quadratic formula?

InvestigationHow High Can You Go?

It’s important to note that a quadratic equation must be in the general form ax2 � bx � c � 0 before you use the quadratic formula.

Solve 3x2 � 5x � 8.

To use the quadratic formula, first write the equation in the form ax2 � bx � c � 0 and identify the coefficients.

3x2 � 5x � 8 � 0

a � 3, b � �5, c � �8

Substitute a, b, and c into the quadratic formula.

x � �b � � ________

b 2 � 4ac _______________ 2a

� �(�5) � �

_______________ (�5)2 � 4(3)(�8) __________________________

2(3)

� 5 � � ____

121 _________ 6

� 5 � 11 ______ 6

x � 5 � 11 ______ 6 � 8 __ 3 or x � 5 � 11 ______ 6 � �1

EXAMPLE BEXAMPLE B

� Solution� Solution


y � �16x2 � 88x � 3

24 � �16x2 � 88x � 3

�16x2 � 88x � 21 � 0; x � 0.25 or x � 5.25. The ball rises to 24 ft on the way up at 0.25 s, and it falls to 24 ft on the way down at 5.25 s.

�16x2 � 88x � 3 � 124; x � 2.75; when �________

b2 � 4ac � 0

�16x2 � 88x � 3 � 200. You get the square root of a negative number, so there are no real solutions.

Step 4 124; the ball reaches the maximum height only once. The ball reaches other heights once on the way up and once on the way down, but the top of the ball’s height, the maximum point, can be reached only once.



Portfolio 10

Group 8

Review 13–17

EXERCISE NOTES

Students practice applying the quadratic formula. Some students will be so intrigued by the ease of using the quadratic formula that they will want to use it with any equation, regard less of its initial form. Once they are comfortable with this method,

encourage them to learn another method, to improve their flexibility.

Exercises 1–3 [Extra Support] Have students compare answers on these exercises and become mindful of common mistakes.

1a. 3x2 � 13x � 10 � 0; a � 3, b � �13, c � �10

1b. x2 � 5x � 13 � 0; a � 1, b � �5, c � �13

1c. 3x2 � 5x � 1 � 0; a � 3, b � 5, c � 1

Discussing the Lesson

LESSON EXAMPLE B

You might ask students to do the calculations on graphing calculators to give them an opportunity to practice inserting parentheses around the numera-tor and around the terms under the radical.

On the calculator screen here, the vertical bar is read as “such that” by the calculator, so “. . .|x � 1” will replace all x’s with the value 1. On the TI-83/84 Plus, you would store 8 _ 3 as x, use a colon, and then type in the original equation: 8/3 → x : 3x2 � 5x � 8.

As the book says, the quadratic formula will solve any quadratic equation, but some solutions will not be real numbers. Students will see nonreal solutions in Lesson 7.5.

� SUPPORT EXAMPLES

1. Use the quadratic formula to find the zeros of the function x2 � 5x � 1. [x 0.209, 4.791]

2. Write the quadratic equation that was used in the quadratic

formula x � �4 � �

_____________ 16�4(�2)(10) _______________ �4 .

� y � �2x2 � 4x � 10�

Closing the Lesson

Restate the quadratic formula and perhaps remind students of how it’s derived from completing the square, with or without using the formulas for h and k.

[Closing Question] “What happens when you apply the quadratic formula to determine where the function y � x2 equals 17?”

ASSIGNING EXERCISES

Suggested Assignments:Standard 1–6, 8, 13

Enriched 4–9, 11, 17

Types of Exercises:Basic 1–5

Essential 1, 4, 5, 6, 8

The solutions are x � 8 _ 3 or x � �1.

To check your work, substitute these values into the original equation. Here’s a way to use your calculator to check.

Remember, you can find exact solutions to some quadratic equations by factoring. However, most quadratic equations don’t factor easily. The quadratic formula can be used to solve any quadratic equation.

EXERCISES

Practice Your Skills

1. Rewrite each equation in general form, ax2 � bx � c � 0. Identify a, b, and c.

a. 3x2 � 13x � 10 b. x2 � 13 � 5x a c. 3x2 � 5x � �1 a

d. 3x2 � 2 � 3x � 0 e. 14(x � 4) � (x � 2) � (x � 2)(x � 4)

2. Evaluate each expression using your calculator. Round your answers to the nearest thousandth.

a. �30 � �

____________ 302 � 4(5)(3) ____________________

2(5) b.

�30 � � ____________

302 � 4(5)(3) ____________________

2(5) a

c. 8 � �

_______________ (�8)2 � 4(1)(�2) _____________________

2(1) d.

8 � � _______________

(�8)2 � 4(1)(�2) _____________________

2(1) a

3. Solve by any method.

a. x2 � 6x � 5 � 0 b. x2 � 7x � 18 � 0 c. 5x2 � 12x � 7 � 0 a

4. Use the roots of the equations in Exercise 3 to write each of these functions in factored form, y � a�x � r1��x � r2�.

a. y � x2 � 6x � 5 b. y � x2 � 7x � 18 c. y � 5x2 � 12x � 7 a

5. Use the quadratic formula to find the zeros of each function.

a. f (x) � 2 x 2 � 7x � 4 b. f (x) � x 2 � 6x � 3 c. y � 6 � �2 x 2

d. 5x � 4 � 2 x 2 � y

Reason and Apply

6. Beth uses the quadratic formula to solve an equation and gets

x � �9 � �

____________ 9 2 � 4(1)(10) ___________________

2(1)

a. Write the quadratic equation Beth started with. a

b. Write the simplified forms of the exact answers.

c. What are the x-intercepts of the graph of this quadratic function?

��

��

You will need

A graphing calculatorfor Exercises 8, 11, and 17.

�0.102

�0.243

�5.898

8.243

x � 1 or x � 5 x � �2 or x � 9 x � �1 or x � �1.4

y � (x � 1)(x � 5) y � (x � 2)(x � 9) y � 5(x � 1)(x � 1.4)

3x2 � 3x � 2 � 0; a � 3, b � �3, c � �2x2 � 15x � 50 � 0; a � 1, b � �15, c � 50

5a. x � 0.5 or x � �4

x � 3 � � __

6 or x � 3 � � __

6 x � � __

3

no real solutions

x2 � 9x � 10 � 0

x � �9 �___

41 __________ 2

�9 � �___

41 __________ 2 and �9 � �

___ 41 __________ 2


Exercise 4c This exercise extends the definition of factored form to include non-integer roots.

Exercise 5 As needed, remind students that each of these problems will have two solutions or one double solution.

Exercise 6 [Language] Simplified form means with the arithmetic performed and the radical reduced, if possible. (In this exercise the radical cannot be reduced.) Use part a to assess how well students understand the variables in the quadratic formula.

[Ask] “Without graphing, can you tell whether the parabola’s vertex is a maximum or a minimum?” [It’s a minimum because a is positive.]

Exercise 7b You might want to ask students to write this equation without a fraction.

7b. y � a(x � 4) � x � 2 __ 5 � or y � a(x � 4)(5x � 2) for a 0

Exercises 8, 9 In discovering what happens when they apply the quadratic formula to an equation with

no solution, students deepen their understanding of the relationships among the terms zeros, x-intercepts, and solutions. They also get a glimpse of the ideas about complex numbers in the next lesson.

8. The solution includes the square root of �36, so there are no real solutions. The graph shows no x-intercepts. Before using the quadratic function, evaluate b2 � 4ac. If b2 � 4ac � 0, then there will be no real solutions.

Exercise 9 This is another opportunity to discuss the discriminant. It can build on conclusions reached in Exercise 8.

10. 1 __ 2 � �b � � ________

b2 � 4ac _______________ 2a �

�b � � ________

b2 � 4ac _______________ 2a

� � 1 __ 2

� �2b ____ 2a � � � b ___ 2a .

The x-coordinate of the vertex,

� b __ 2a , is midway between the two

x-intercepts.

Exercise 11 This exercise requires using finite differences and the quadratic formula. [Alert] The plug was pulled at time 0, not time 1.

Exercise 12 The equation can be solved by the quadratic formula. Only the positive solution,

1 � � __

5 ______ 2 , makes sense as a length.

[Context] History Connection Students might be interested to know how the golden ratio and Fibonacci sequence are related. In the Fibonacci sequence (1, 1, 2, 3, 5, 8, . . . , 4181, 6765, 10946, . . .) each term is the sum of the two preceding terms. The ratio of consecutive numbers in the sequence approaches the golden ratio.

10946 _____ 6765 1.618033999

1 � � __

5 _______ 2 1.618033989

Ratios of larger Fibonacci numbers will be even closer to the golden ratio.

7. Write a quadratic function whose graph has these x-intercepts.

a. 3 and �3 a b. 4 and ��25� c. r1 and r2

8. Use the quadratic formula to find the zeros of y � 2x2 � 2x � 5. Explain what happens. Graph y � 2x2 � 2x � 5 to confirm your observation. How can you recognize this situation before using the quadratic formula?

9. Write a quadratic function that has no x-intercepts.

10. Show that the mean of the two solutions provided by the quadratic formula is � b __ 2a . Explain what this tells you about a graph.

11. These data give the amount of water in a draining bathtub and the amount of time after the plug was pulled.

a. Write a function that gives the amount of water as a function of time. a

b. How much water was in the tub when the plug was pulled?

c. How long did it take the tub to empty?

12. A golden rectangle is a rectangle that can be divided into a square and another smaller rectangle that is similar to the original rectangle. In the figure at right, ABCD is a golden rectangle because it can be divided into square ABFE and rectangle FCDE, and FCDE is similar to ABCD. Setting up a proportion of the side lengths of the similar rectangles leads to a

____ a � b � b _ a . Let b � 1 and solve this equation for a.

Review

13. Complete each equation.

a. x2 � ? � � 49 � (x � ? � )2 a b. x2 � 10x � ? � � ( ? � )2 a

c. x2 � 3x � ? � � ( ? � )2 d. 2x2 � ? � � 8 � 2(x2 � ? � � ? � ) � ? � (x � ? � )2

14. Find the inverse of each function. (The inverse does not need to be a function.)

a. y � (x � 1)2 b. y � (x � 1)2 � 4 c. y � x2 � 2x � 5 a

��

B C

A

a b

a

F

E D

The Fibonacci Fountain in Bowie, Maryland, was designed by mathematician Helaman Ferguson using Fibonacci numbers and the golden ratio. It has 14 water spouts arranged horizontally at intervals proportional to Fibonacci numbers.

History

Many people and cultures throughout history have felt that the golden rectangle is one of the most visually pleasing geometric shapes. It was used in the architectural designs of the Cathedral of Notre Dame in Paris, as well as in music and famous works of art. It is believed that the early Egyptians knew the value of the golden ratio (the ratio of the length of the golden rectangle

to the width) to be 1 � � __

5 _____ 2 and that they used the ratio when building their

pyramids, temples, and tombs.

Time (min)x 1 1.5 2 2.5

Amount of water (L)y 38.4 30.0 19.6 7.2


y � � __

x � 1 y � �_____

x � 4 � 1 y � �_____

x � 6 � 1

y � a(x � 3)(x � 3) for a 0 y � a(x � r1)(x � r2) for a 0

The function can be any quadratic function for which b2 � 4ac is negative. Sample answer: y � x2 � x � 1.

y � �4x 2 � 6.8x � 49.2

49.2 L

2.76 min

a _____ a � 1 � 1 __ a , a � 1 � __

5 _______ 2

7.2

5.5

13a. x2 � 14x � 49 � (x � 7)2 or x2 � (�14x) � 49 � [x � (�7)]2

13b. x2 � 10x � 25 � (x � 5)2

x2 � 3x � 9 __ 4 � � x � 3 __ 2 � 2

13d. 2x2 � 8x � 8 � 2(x2 � 4x � 4) � 2(x � 2)2 or 2x2 � (�8x) � 8 � 2[x2 � (�4x) � 4] � 2[x � (�2)]2



Exercise 16 Students may need to sketch a diagram to figure out the equation using the Pythagorean Theorem. Assume that the wall is vertical and the ground is horizontal.

Exercise 17 Although there are 11 support cables labeled, their distance apart is 1 __ 12 of 160 ft, or 13 1 _ 3 ft. Students may correctly reason that a bridge would have two parabolic cables, one on each side of the road. Their answer of 458.3 ft is correct for the assumption they are making.

17. a � k � 52.08 _

3 ft; b � j � 33.

_ 3 ft; c � i � 18.75 ft; d �

h � 8. _

3 ft; e � g � 2.08 _

3 ft; f � 0; total length � 229.1

_ 6 ft

[Context] Engineering Connection Students who completed the Chapter 5 exploration The Number e might be interested to know that the equation for a catenary curve contains the number e.

y � a __ 2 � e x/a � e�x/a �

The graph of this function is symmetric about the y-axis; a is the y-intercept.

EXTENSION

Have students use geometry soft-ware to analyze what happens to the vertex of a parabola when you vary a in the equation of the gen-eral form but hold b and c con-stant. Students then can describe what happens when b varies and a and c are held constant. This could also be done with Fathom or the TI-Nspire.

If you don’t want to challenge students to devise their own programs, you can give them Calculator Note 7D, which contains a sample program.

� The report includes a simple program that gives the solutions as decimal approximations.

� Documentation is provided to confirm that the program works.

� The student has created a program that can give solutions as exact values in the form of reduced radical expressions.

� The student has created a program that checks the discriminant to tell the user when the number of roots is 0, 1, or 2.

OUTCOMESSupporting the

15. Convert these quadratic functions to general form.

a. y � (x � 3)(2x � 5) b. y � �2(x � 1)2 � 4

16. A 20 ft ladder leans against a building. Let x represent the distance between the building and the foot of the ladder, and let y represent the height the ladder reaches on the building.

a. Write an equation for y in terms of x. a

b. Find the height the ladder reaches on the building if the foot of the ladder is 10 ft from the building. a

c. Find the distance of the foot of the ladder from the building if the ladder must reach 18 ft up the wall.

17. APPLICATION The main cables of a suspension bridge typically hang in the shape of parallel parabolas on both sides of the roadway. The vertical support cables, labeled a–k, are equally spaced, and the center of the parabolic cable touches the roadway at f. If this bridge has a span of 160 ft between towers, and the towers reach a height of 75 ft above the road, what is the length of each support cable, a–k? What is the total length of vertical support cable needed for the portion of the bridge between the two towers?

y

x

20 ft

Engineering

The roadway of a suspension bridge is suspended, or hangs, from large steel support cables. By itself, a cable hangs in the shape of a catenary curve. However, with the weight of a roadway attached, the curvature changes, and the cable hangs in a parabolic curve. It is important for engineers to ensure that cables are the correct lengths to make a level roadway.

A chain hangs in the shape of a catenary curve.

ab

cd e

fg h

ij

k

CALCULATOR PROGRAM FOR THE QUADRATIC FORMULA

Write a calculator program that uses the quadratic formula to solve equations. The program should calculate and display the two solutions for a quadratic equation in the form a x 2 � bx � c. Depending on the type of calculator you have, the user can give the a, b, and c values as parameters for the program (as shown) or the program can prompt the user for those values.

Your project should include� A written record of the steps your program uses.� An explanation of how the program works.� The results of solving at least two equations by hand and with your program to verify

that your program works.

quad(1,-1,-6)

3

-2

Done

y � �2x2 � 4x � 2

y � �________

400 � x2

approximately 17.32 ft

approximately 8.72 ft

y � 2x2 � x � 15

7.2


the quadratic formula 7 - high school math | prek...

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