Lesson 6
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| Course: | Math 20-2 SS |
| Book: | Lesson 6 |
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| Date: | Saturday, 6 September 2025, 2:32 AM |
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1. Lesson 6
Module 3: Quadratics
Lesson 6: Binomial and Trinomial Factoring Review
Focus
The path of a soccer ball after it is kicked can be modelled with a quadratic function. How do you get the kickoff point and landing point from the function? How do zeros of the function relate to the solution of a related equation?
© Christian Kieffer/12734687/Fotolia
In this lesson 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:
- What processes can be used to fully factor binomials and trinomials?
- How can you use a quadratic equation in factored form to find the roots of the quadratic equation?
- How can you explain the relationships among the roots of an equation, the zeros of the corresponding function, and the x-intercepts of the graph of the function?
Assessment
There is no formal assessment for this lesson. You need to ensure you are comfortable with all methods of factoring in preparation for further application questions.
1.1. Launch
Module 3: Quadratics
Launch
Before beginning this lesson, you should be able to
- find common factors
- factor trinomials with a leading coefficient of 1
1.2. Are You Ready?
Module 3: Quadratics
Are You Ready?
Factor the following trinomials.
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 concepts before continuing through the lesson.
1.3. Refresher
Module 3: Quadratics
Refresher
If you experienced difficulties with the questions, view the video “Factoring Trinomials with a Leading 1 Coefficient” showing common factoring and factoring a trinomial with a leading coefficient of 1 in one question.
Khan Academy (CC BY-NC-SA 3.0)
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
Now that you have reviewed factoring trinomials with a leading coefficient of 1, you are ready to review the factoring of trinomials with other leading coefficients.
Try This 1
Examine the following trinomials. Place the trinomials you think will be easy to factor in one list, and place those that you feel will be more difficult to factor in the other list.
- 10x2 − 100x + 160
- 3x2 − 21x − 24
- 7x2 + 3x + 11
- 2x2 − 6x + 4
- 6x2 + x − 15
- 12x2 − 5x − 3
© Bombaert Patrick/27091219/Fotolia
As you are creating your lists, think about the reasons why you are putting each trinomial in either list.
Share 1
Share your lists from Try This 1 with a partner. Discuss why you chose to put each trinomial in your lists. Were your lists and your partner’s lists the same? Were your reasons the same?
Watch the “Factoring Trinomials with a Common Factor” video to see why some trinomials could be easier to factor than others.
Khan Academy (CC BY-NC-SA 3.0)
The video showed that having a common factor is a simplifying feature. These trinomials are not much harder to work with than trinomials with a leading coefficient of 1. Try working with a few of them.
Self-Check 1
Factor each of the following trinomials.
1.5. Explore
Module 3: Quadratics
Explore
There are many other ways to approach the factoring of trinomials. The video “Factoring Trinomials with a non-1 Leading Coefficient by Grouping” shows one way to factor trinomials that may have been on your hard-to-factor list in the Discover section.
Khan Academy (CC BY-NC-SA 3.0)
Self-Check 2
Factor each of the following trinomials.
- 8x2 + 14x + 3 Answer
- 3x2 + 5x − 2 Answer
- 6x2 + x − 15 Answer
- 12x2 − 5x − 3 Answer
- 3x2 + 20x + 12 Answer
In mathematics, some trinomials are classified as perfect squares. These trinomials have two identical factors. Practice is needed to decide when a trinomial is a perfect square. Watch Factoring Perfect Square Trinomials to learn more.
Self-Check 3
© komar.maria/31328146/Fotolia
Factor each of the following trinomials.
- x2 − 4x + 4 Answer
- x2 + 10x + 25 Answer
- y2 − 6y + 9 Answer
- 2x2 + 24x + 72 Answer
- 4x2 − 20x + 25 Answer
1.6. Explore 2
Module 3: Quadratics
Difference of Squares
Some factoring questions will only have two terms. The process used to factor in such cases is called factoring by a difference of squares. To factor by a difference of squares, a question must have two terms and a subtraction sign between the terms, and each term must be a perfect square.
Watch Factoring a Difference of Squares to learn more.
Self-Check 4
Factor the following binomials.
You have practised your factoring skills in preparation for solving quadratic equations. The factoring questions you have been working on can be changed into quadratic equations by making the questions equal to 0.
For example, if you were asked to factor x2 + x − 20, you would get the answer (x + 5)(x − 4). If you were asked to solve x2 + x − 20 = 0, you would get (x + 5)(x − 4) = 0 and then proceed to the roots x = −5 and x = +4. The sign will always be reversed because x + 5 = 0 only when x = −5 and x − 4 = 0 only when x = +4.
If there is a factor like (2x − 3), you can start by setting the factor equal to 0.
2x − 3 = 0
The variable needs to be isolated, so the 3 needs to be moved to the other side.
2x = 3
Then divide by 2 to get 
This is a rational root, since it is a rational number.
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. You might choose to add the following terms to your copy of Glossary Terms:
- perfect square
- rational root
- integral root
1.7. Explore 3
Module 3: Quadratics
Try This 2
Open “Solving Quadratic Equations: Factoring.” Complete all parts of the lesson. Use the arrows to navigate between screens. Press Enter on your keyboard once you have entered an answer.
How are quadratic functions and quadratic equations related?
Did You Know?
The quadratic function f(x) = ax2 + bx + c can be changed to a quadratic equation by replacing f(x) with 0 to get 0 = ax2 + bx + c.
Since f(x) is another symbol for y, you are replacing y with 0 and finding the x-intercepts. Therefore, 0 = ax2 + bx + c, or ax2 + bx + c = 0, is considered the standard form of a quadratic equation.
You studied quadratic functions in Lessons 1 to 5. You are now starting to work with quadratic equations. As you have seen, solving quadratic equations gives you the roots of the equation. These roots are the x-intercepts or zeros of the related quadratic function.
Self-Check 5
© Sylvie Bouchard/34161049/Fotolia
A header in soccer is not usually aimed at getting the maximum distance. Instead, the ball is being directed to go to a particular place. This allows the player who is heading the ball or the player’s teammate to generate a scoring opportunity.
Suppose the height of a header is given as a function of time, h = −4.9t2 +19.6t, where the time is in seconds and the height is in metres. If the ball is grabbed by the goalkeeper just before entering the net, how long will the ball be in the air?
Assume that the height of the ball when caught is the same as the height when it left the player’s head and that the coordinate axes system has its origin at the player’s head. Solve by factoring.
Remember that roots refer only to equations—in this case, quadratic equations.
- The roots of a quadratic equation have the same value as the x-intercepts of the graph of the corresponding quadratic function.
- The roots of the equation have the same value as the zeros of the corresponding quadratic function.
If you haven’t done so already, you may want to add the factoring methods you studied in this lesson to your notes organizer.
If you feel you need a bit more practice, complete Self-Check 6.
Self-Check 6
1.8. Lesson 6 Summary
Module 3: Quadratics
Lesson 6 Summary
In this lesson you investigated the following questions:
- What processes can be used to fully factor binomials and trinomials?
- How can you use a quadratic equation in factored form to find the roots of the quadratic equation?
- How can you explain the relationships among the roots of an equation, the zeros of the corresponding function, and the x-intercepts of the graph of the function?
You learned that the roots of a quadratic equation are the solutions to the corresponding function set 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 those functions.
In the next few lessons you will learn other strategies for solving quadratic equations using graphs and the quadratic formula.