Lesson 8
| Site: | MoodleHUB.ca 🍁 |
| Course: | Math 20-2 SS |
| Book: | Lesson 8 |
| Printed by: | Guest user |
| Date: | Saturday, 6 September 2025, 2:39 AM |
Description
Created by IMSreader
1. Lesson 8
Module 3: Quadratics
Lesson 8: Solving Quadratic Equations Using the Quadratic Formula
Focus
Jupiterimages/Comstock/Thinkstock
When this snowboarder takes to the air, she is not thinking about how her jump could be described by quadratic functions. She is trying to maximize her air time so she can grab the board, a neat trick. To improve her jump, an analysis using quadratic functions might be useful. She could see that her vertical liftoff speed is the determining factor in how long she remains aloft.
In this lesson you will use formulas to find the roots of quadratic functions and analyze parabolic paths.
Lesson Questions
This lesson will help you answer the following inquiry questions:
- How can you solve a quadratic equation without using factoring or graphing?
- How can you use the discriminant to find the number of roots a quadratic function will have?
- How do you solve problems using the quadratic formula?
Assessment
- Share 1
- Lesson 8 Assignment
All assessment items you encounter need to be placed in your course folder.
Save a copy of the Lesson 8 Assignment to your course folder. You will receive more information about how to complete the assignment later in this lesson.
1.1. Launch
Module 3: Quadratics
Launch
This section presents questions to help you determine if you have the skills and knowledge to complete this lesson successfully. Before beginning this lesson, you should be able to determine the square root of a number or an expression by first simplifying the expression.
1.2. Are You Ready?
Module 3: Quadratics
Are You Ready?
Simplify and find the square root of the following expressions.
1. 
Answer
2. 
Answer
3. 
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 resource in the Refresher section to review the concepts before continuing through the lesson.
1.3. Refresher
Module 3: Quadratics
Refresher
Go to Square Root to read a definition and work through an applet.
Go back to the Are You Ready? section, and try the questions again. If you are still having difficulty, contact your teacher.
1.4. Discover
Module 3: Quadratics
Discover
© treenabeena/13191894/Fotolia
It is often easiest to factor when solving a quadratic equation, ax2 + bx + c = 0. You have used graphing technology to find the roots of equations that are not factorable. There is a third way to find the roots of a quadratic equation. You can use a formula called the quadratic formula. This formula always works, whether the original equation is factorable or not.
The quadratic formula, 
, is a quick and efficient way to solve quadratic equations. The formula was discovered and has been used by many cultures going back well before 300 BCE.
Quadratic Formula Proof shows the derivation of the quadratic formula.
Watch Using the Quadratic Formula to see an example of using the quadratic formula to solve an equation that is not factorable.
Study “Example 2: Connecting the quadratic formula to factoring” on page 416 of the textbook. Pay attention to how the values for a, b, and c were substituted into the formula. Also, look closely at the note about the discriminant being a perfect square. This might be useful in some factoring questions.
Self-Check 1
Solve the equation 15x2 − 6 = x using the quadratic formula. Answer
1.5. Explore
Module 3: Quadratics
Explore
© flucas/968629/Fotolia
While you’ve been working with the quadratic formula, you had to evaluate b2 − 4ac. “Example 2” on page 416 of the textbook noted that if b2 − 4ac was a perfect square, the associated quadratic function, y = ax2 + bx + c, could be factored. The expression b2 − 4ac, which is known as the discriminant, can tell you more about the associated quadratic function.
The following applet allows you to explore how values of the discriminant and the graph of a quadratic function are related.
Try This 1
Open to the Quadratic Function: Discriminant applet to explore how the discriminant is related to the number of x-intercepts.
The applet has sliders to change the values of the constants (a, b, and c) in y = ax2 + bx + c. Notice the information given below the sliders. You should see the discriminant, b2 − 4ac, and its value of 12 when initial values of a = 1, b = 0, and c = −3 are used.
Move the b-slider or the c-slider. What happens to the discriminant? What happens to the x-intercepts?
Experiment with the sliders and notice the values of the discriminant when there
- are two x-intercepts
- is one x-intercept
- are no x-intercepts
- What is the relationship between the value of the discriminant and the number of x-intercepts?
- How does each situation relate to the position of the vertex?
- Do your generalizations also hold true when the a-value is negative? Move the a-slider and see. What do you conclude?
Share 1
Submit your work for Share 1 to your teacher for marking.
Based on your observations from Try This 1, discuss the following questions with another student or appropriate partner.
- How can the value of the discriminant tell you whether there will be one, two, or zero x-intercepts? (3 marks)
- Does your answer depend on whether the value of a is negative or positive? Why or why not? (2 marks)
- Summarize your discussion by creating a general rule relating the value of the discriminant to the number of x-intercepts. (3 marks)
Place a copy of your responses in your course folder for reference later in the course.
1.6. Explore 2
Module 3: Quadratics
Recall from the Course Introduction that you will be creating your own course glossary. Open the Glossary Terms document that you saved to your course folder, and add in any new terms. It is suggested that you add the following new terms to your Glossary Terms document:
- discriminant
- quadratic formula
Self-Check 2
Complete “Check Your Understanding” questions 1.a., 1.b., 2.c., 2.d., 3.c., and 3.d. on page 419 of the textbook to practise using the quadratic formula before moving on in this lesson. You might want to compare your answers to the answers provided after responding to each question. Answer
Solving quadratic equations can be used in many problem situations. Watch the piece titled Flying Snowboard.
Photo: Jupiterimages/Comstock/Thinkstock
Self-Check 3
© Ilja Mašík/11361648/Fotolia
A snowboarder takes a jump and lands down the slope. His approximate height above the landing is h = −5x2 + 3x + 3.6, where x represents time in seconds.
- Calculate the discriminant and identify how many roots there will be. Answer
- How long was the snowboarder in the air? Answer
- What is the maximum height during the jump? Round your answer to two decimal places. (You might like to use the Quadratic Function: Discriminant applet from the Explore section to complete part of this solution). Answer
The quadratic formula allows you to find the roots of an equation, whether the equation is factorable or not, as long as the discriminant is greater than or equal to zero. If the discriminant is a perfect square, the function is factorable.
Self-Check 4
- An outdoor clothing retailer is overstocked with one style of snowboarding jacket. In order to sell more of the jackezts, the store has decided to discount the price. The marketing manager has figured out a quadratic function describing the net amount of profit, P, the store will earn from the sales.
The function is P = −30x2 + 350x + 5000. In this function, x represents the number of successive $5 discounts to the price.
© BEAUTYofLIFE
/17190056/Fotolia
- Assume you own 30 apartment units. When the rent is $800 per month, all the apartments are occupied. For each rent increase of $40 per month, one of the units becomes unoccupied.
The following function relates the price increase, p, to the revenue, r.
What rent should you charge to ensure you earn $25 000? Answer
Look at the “Frequently Asked Questions” on page 423 of the textbook.
If you haven’t done so already, you may want to add what you’ve learned about the quadratic formula and the discriminant to your notes organizer.
If you feel you need a bit more practice, complete Self-Check 5 before moving to Connect.
Self-Check 5
1.8. Lesson 8 Summary
Module 3: Quadratics
Lesson 8 Summary
© Melissa Schalke/7021195/Fotolia
In this lesson you investigated the following questions:
- How can you solve a quadratic equation without using factoring or graphing?
- How can you use the discriminant to find the number of roots a quadratic function will have?
- How do you solve problems using the quadratic formula?
You found that the discriminant, b2 − 4ac, can give you information about the quadratic function y = ax2 + bx + c.
- If the discriminant equals zero, there will be one root or one x-intercept on the graph.
- If the discriminant is positive, there will be two roots or two x-intercepts on the graph.
- If the discriminant is negative, there will be no roots or no x-intercepts on the graph.
- If the discriminant is a perfect square, the function can be factored.
You worked with the quadratic formula, 
, using the formula to find roots of quadratic equations in the form 0 = ax2 + bx + c. For some problems, only one of the roots is admissible because the other does not make sense in the situation.
You learned to apply the quadratic formula to various scenarios to solve problems. You calculated how long a snowboarder was in the air and how far and how high the boarder jumped. You were also able to answer questions about maximizing profits.
In the last lesson in Module 3 you will practise applying many of the skills you learned to practical situations.