Lesson 7

Lesson 7: Solving Quadratic Equations by Factoring or Graphing

 

Focus

 

Parabolic reflectors, whose shape can be modelled by quadratic functions, often receive radio waves from outer space. In the Very Large Array, 27 radio antennas on rails sit on the Plains of San Agustin, 50 miles west of Socorro, New Mexico. Each antenna is 25 m (the height of an eight-storey building) in diameter and weighs 209 t (tonnes). The data from the antennas is combined electronically to give the resolution of an antenna 36 km across and the sensitivity of a dish 130 m in diameter.

 

As you progress through this lesson, you will analyze parabolic shapes in sports and communications. You will learn how to use the zeros of a quadratic function to solve corresponding equations and answer related problems.

 

Lesson Questions

 

This lesson will help you answer the following inquiry questions:

• How can you use a quadratic equation in factored form to solve problems involving the zeros of the corresponding quadratic function or the x-intercepts of the graph of the function?
  
  
• How can you find the roots of a quadratic equation by graphing the corresponding quadratic function?

Assessment

• Share 1
• Lesson 7 Assignment

All assessment items you encounter need to be placed in your course folder.

 

Save a copy of the Lesson 7 Assignment to your course folder. You will receive more information about how to complete the assignment later in this lesson.

 

Materials and Equipment

• graphing calculator

Launch

 

Before beginning this lesson, you should be able to

• factor and solve quadratic equations
  
  
• graph quadratic functions and find zeros using your graphing calculator

Are You Ready?

1. Solve each of the following quadratic equations using factoring.
   
   
   1. x2 + 5x + 4 = 0 Answer
      
      
   2. 2x2 + 4x − 30 = 0 Answer
      
      
   3. y2 − 16 = 0 Answer
      
      
   4. −5y2 + 125 = 0 Answer
      
      
   5. 2x2 + 5x − 3 = 0 Answer
      
      
2. Using your calculator, graph and find the zeros of the following quadratic functions.
   
   
   1. f(x) = x2 − x − 6 Answer
      
      
   2. f(x) = 3x2 + x − 2 Answer
      
      
   3. f(x) = x2 + 6x + 9 Answer

If you successfully completed the Are You Ready? questions, move on to the Discover section of the lesson.

 

If you experienced difficulties with the questions, use the resources in the Refresher section to review the concepts before continuing through the lesson.

 

Refresher

 

Review the following examples to see how to factor quadratic equations. Once you have reviewed some examples, try some questions to make sure you understand the concepts provided here.

 

Example 1

 

x2 + 5x + 4 = 0

 

Since 1 and 4 have a sum of 5 and a product of 4, x2 + 5x + 4 = 0 is the same as

 

 

Example 2

 

x2 − 9x + 20 = 0

 

Since −4 and −5 have a sum of −9 and a product of 20, x2 − 9x + 20 = 0 is the same as

 

 

Example 3

 

x2 − 4x − 32 = 0

 

Since 4 and −8 have a sum of −4 and a product of −32, x2 − 4x − 32 = 0 is the same as

 

 

Example 4

 

2x2 + 4x − 30 = 0

 

The equation has a common factor of 2.

 

2(x2 + 2x − 15) = 0

 

Since −3 and 5 have a sum of 2 and product of −15, 2(x2 + 2x − 15) = 0 is the same as

 

 

Example 5

 

y2 − 16 = 0

 

Since both terms on the left side are squares, this is a difference of squares (16 = 42).

 

 

Example 6

 

−5y2 + 125 = 0

 

The equation has a common factor of −5.

 

−5(y2 − 25) = 0

 

Since both terms inside the brackets are squares, treat this like a difference of squares.

 

 

Example 7

 

x2 − 8x + 16 = 0

 

The equation has the form of a perfect square since −8x = 2(−4x) and 16 = (−4)2.

 

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Now that you have had an opportunity to review some examples, solve the following factoring questions to ensure you understand the concepts presented here.

1. 2x2 + 8x + 8 = 0 Answer
   
   
2. 2x2 + 5x − 3 = 0 Answer
   
   
3. 4x2 + 5x − 6 = 0 Answer

Go to How to Find Zeros, Minimums, and Maximums with Your Graphing Calculator to see directions for finding the zeros of a function for a graphing calculator. You may have to adapt the directions if you are using a calculator from a different manufacturer.

 

Go back to the Are You Ready? section, and try the questions again. If you are still having difficulty, contact your teacher.

 

Discover

 

Try This 1 gives you a chance to play with the factored form of the quadratic function. The activity will let you see how the various parameters affect the shape of the function’s graph.

 

Try This 1

 

Use Polynomial Function Explorer to explore the factored form of the quadratic equation. Change the “Degree =” slider to 2 so you get a second-degree polynomial—a quadratic function. Choose “Show x-intercepts” by clicking on the square. Experiment with the b-slider to see how the slider changes the values in the factored form of the function, the intercepts, and the position of the graph.

 

1. When you are satisfied that you know what the b-slider does and how changing the value affects the graph, set the value of b to 4. Experiment with the k-slider to see how the value of k changes the function, the intercepts, and the position of the graph. How many different graphs are possible that have x-intercepts of 1 and 4?
   
   
2. If you put k = 2, the graph passes through (5, 8). Can you make another graph with the x-intercepts of 1 and 4 that passes through (5, 8)? Move the k-slider and see if you can find one. What can you conclude about a graph where the x-intercepts and one other point are defined?

Share 1

 

Explore

 

You’ve just discovered some facts about quadratic functions. You’re now going to use this information and your knowledge of factoring to solve quadratic equation questions.

 

As you saw in Discover, it is straightforward to find the x-intercepts of a quadratic function in factored form. Is it as simple to find the factored form of a quadratic given its x-intercepts?

 

Suppose you are given a quadratic function with x-intercepts of x = −1 and x = 3. Write two quadratic functions that match this information.

 

Since you know that x = −1 and x = 3 are x-intercepts, you can find the (x − r) and (x − s) factors of f(x) = a(x − r)(x − s) quite easily.

 

 

The equations show that f(x) = a(x + 1)(x − 3) has the required x-intercepts.

 

To find more solutions, you can change the value of a. Options here are endless. You might choose 2 or or −3, giving the following functions with the same x-intercepts:

 

f(x) = 2(x2 − 2x − 3)

 

f(x) = (x2 − 2x − 3)

 

f(x) = −3(x2 − 2x − 3 )

 

The first two functions open upwards, and the third function opens downwards.

 

Self-Check 1

 

Suppose the height of a kicked ball is given as a function of distance, h = −0.0218d2 + 1.308d, where the distance and the height are in metres. How far will the ball go if it hits the ground before being caught? Answer

 

Manipulating Equations to Equal Zero

 

Quadratic equations may not always have one side equal to 0 (e.g., −6 = x2 + 5x). If you are working with a quadratic equation like this, you can manipulate the equation to get all non-zero terms on one side and 0 on the other side. Then you can use an appropriate factoring strategy to solve the equation.

 

For example, if you were asked to solve −6 = x2 + 5x, you might proceed as follows.

 

To move −6, add 6 to both sides.

 

 

You can now factor the right side of the equation.

 

 

Now you can find values for x easily.

 

So, x = −2 and x = −3 are the required solutions.

 

Self-Check 2

 

Solving Equations Using Technology

 

In the next few activities you will use a graphing calculator or graphing technology to find the intercepts of a quadratic function or points of intersection of two functions.

Connect

 

Lesson Assignment

 

Lesson 7 Summary

 

In this lesson you investigated the following questions:

• How can you use a quadratic equation in factored form to solve problems involving the zeros of the corresponding quadratic function or the x-intercepts of the graph of the function?

• How can you find the roots of a quadratic equation by graphing the corresponding quadratic function?

You found that there could be an infinite number of quadratic functions that have the same zeros or x-intercepts. Among those functions, a certain function could be specified by giving the coordinates of a third point on the parabola or by giving the value of a in the function y = a(x − r)(x − s).

 

You also learned that the roots of a quadratic equation are the solutions to the corresponding function equal to zero. The roots have the same values as the x-intercepts of the graph of the corresponding function and the zeros of that function.

 

By changing a function into a corresponding equation with one side equal to zero, you were able to solve for the x-intercepts of the graphs and the zeros of the function. This enabled you to answer questions involving the functions.

 

You learned how to find the x-intercepts of a function using a graphing calculator. You practised using the zero capability of the CALCULATE feature to answer problems involving a variety of situations.

 

In Lesson 8 you will learn a new process for solving quadratic equations to find their roots.