Project AMP
A. Barnes, Drake, Graham, Sharp,
Dr. Antonio R. Quesada, Director Project AMP
Lesson Title: Introduction to Solving Quadratic Equations by Factoring
Lesson Summary:
This lesson will lead students through the steps needed to solve quadratic equations of type
x 2  bx  c  0 by factoring. Students will “warm up” by solving simple quadratic equations
algebraically, graphically, and using the table feature of a graphing calculator. The lesson will then
lead students step-by-step from solving a quadratic equation by graphing to solving one by
factoring. The lesson will introduce students to quadratic equations of type x 2  bx  c  0 .
Key Words: quadratic equation, factoring, zeros, roots
Background Knowledge:
Students will have learned how to solve and graph linear equations.
Students will have learned terminology, such as intercepts, roots, zeros.
Students will have learned how to graph an equation on a graphing calculator.
Students should be familiar with standard form of quadratic equations
Ohio Grade Level Indicators: Patterns, Functions, and Algebra
Grade 8, #12. Solve simple quadratic equations graphically; e.g., y = x2 – 16.
Grade 9, #10: Solve quadratic equations with real roots by factoring, graphing, using the
quadratic formula and with technology.
Grade 10, #10: Solve real-world problems that can be modeled using linear, quadratic,
exponential or square root functions.
Learning Objective: To solve quadratic equations by factoring and to understand what the solutions
represent in a problem situation.
Materials: Student packet “Solving Quadratics by Factoring” and a graphing calculator.
Pictures of real world parabolas (Samples appear on the last pages.)
Procedure: 1. Show pictures of real world parabolas. Discuss how they model real world quadratic
equations. Depth of discussion depends on class level and type. Pictures are at the bottom
of the lesson plan page.
2. Review solving linear equations, as needed.
3. Break into small student groups.
4. Distribute the student packet.
5. Explain that the packet should lead the students to form some conclusions about how to
solve quadratic equations by factoring.
6. Distribute graphing calculators as needed.
7. Walk around and monitor student progress.
8. Debrief the packet.
Assessment(s): Student packet “Solving Quadratics by Factoring.”
Quadratic equations will also be included on the unit quiz/test.
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Dr. Antonio R. Quesada, Director Project AMP
http://www.pbs.org/wgbh/amex/goldengate/
http://www.carondelet.pvt.k12.ca.us/Family/Math/03210/page3.htm
http://en.wikipedia.org/wiki/Image:Double-alaskan-rainbow.jpg
http://en.wikipedia.org/wiki/Hoover_Dam
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Dr. Antonio R. Quesada, Director Project AMP
http://www.glenbrook.k12.il.us/gbssci/Phys/Class/vectors/u3l2a.html
Project AMP
Dr. Antonio R. Quesada, Director Project AMP
Solving Quadratics by Factoring
Based on our previous work, we know that we can find solutions to quadratic equations graphically by
first setting the equation equal to zero and then finding the x-intercepts of the graph. Our goal for this
investigation is to be able to solve quadratic equations by factoring.
1. Solve the following equation algebraically and check your solution graphically and using a table.
a. x 2  8  17
Graphically
Table
X
Y
0
0
0
b. x 2  16  33
Graphically
Table
X
Y
0
0
0
c. x 2  11  47
Graphically
Table
X
0
Y
0
0
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Dr. Antonio R. Quesada, Director Project AMP
2. Do your solutions to each equation match for all three of your methods. Explain why or not.
3. Generalize and explain how your three solution methods are related to each other.
4. Solve the following quadratic equations algebraically. Check them by using one of the other two
methods. You will need to enlarge your viewing window to check some of the graphs.
a. x 2  49 _____________ b. x 2  20
_______________ c. x 2  14  50 ______________
5. Solve the following using the given method.
a. 10  x 2  2 x  5 __________ b.  8  2 x 2  6 x __________ c. 56  3x 2  8 __________
Solve by using a table.
Solve by graphing.
Solve using algebraic steps.
When graphing a parabola, the points where the graph crosses the x-axis are called the zeroes, or roots,
of the parabola. These are the points where y = 0.
For example, to find the roots of the graph for y  x 2  9 we could make y = 0 and use the graph. Try it.
We could also set y = 0 and solve by using algebraic steps. Show work:
6. Now, find the roots for y  x 2  2 x  15 using the graph. x = _______ and x = ________
What happens when we try to solve using the algebraic steps we’ve used in the past?
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Dr. Antonio R. Quesada, Director Project AMP
Recall that standard form for quadratic functions is y  ax 2  bx  c . We will have to use a new
algebraic process to solve quadratics with a b-coefficient. One of the possible methods is called factoring
the quadratic. This involves re-writing a quadratic equation in terms of its factors. Factors are terms or
quantities that are multiplied. So, in order to understand factoring, we first need to review multiplication
of algebraic terms or quantities using the distributive property (FOIL).
7. Multiply the following factors to rewrite the expression in standard form using the distributive
property. Remember to combine the like terms after you multiply.
a. (x  3)(x + 5)
b. (x + 2)(x  6)
c. (x  7)(x  3)
d. (x + 4)(x  4)
8. What pattern do you notice in regard to the b-coefficient? Hint: It will involve a sum.
9. What pattern do you notice in regard to the constant c-value? Hint: It will involve a product.
You should have noticed: The middle coefficient (b) is the sum of the two numbers in the factors.
The constant term (c) is the product of the two numbers in the factors.
10. Now, let’s rewrite the following standard form expressions into their factored form.
a. x 2  2 x  15
b. x 2  4 x  12
c. x 2  7 x  10
d. x 2  36
11. Let’s try some more factoring. Write the following in their factored form.
a. x 2  x  6
b. x 2  9 x  18
c. x 2  6 x  8
d. x 2  64
e. x 2  4 x  21
f. x 2  6 x  9
g. x 2  16 x  64
h. x 2  x  30
12. Graph #11a, #11b, and #11e. Write down the roots from each graph. Explain how these roots are
related to their factored forms.
#11a roots: _________, #11b roots ________, and #11e roots _________
Explain:
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Dr. Antonio R. Quesada, Director Project AMP
13. So, using your results from question #11 and 12, give the values of x that would make the
expressions equal to zero.
#11a x-values: ______________, #11b x-values ____________, and #11e x-values ____________
14. Generalize your results from questions #11, #12, and #13. How are factors, roots, and zeroes
related to each other?
15. By making the expressions equal to zero, we have formed equations. What is another term that we
can use for these x-values that make the equation equal to zero?
So how does factoring help us? Since we can write equations into factored form, we more easily find the
roots or zeroes to help us solve equations. These are called the solutions to the quadratic equation. For
example, let’s go back and solve x 2  2 x  15  0 using algebraic steps by factoring.
If we rewrite x 2  2 x  15  0 into factored form we get (x  3)(x + 5) = 0. One solution to the equation
is x = 3 because if we substitute 3 into the equation for x, we would get (3  3)(3 + 5) or (0)(8), which
gives us 0. The other solution is x = 5 because if we substitute 5 into the equation for x, we would get
(5  3)(  5 + 5) or (8)(0), which also gives us 0. So, the two values of x that would solve the equation
are x = 3 and x = 5. We can of course verify this by looking at the graph and finding the zeroes.
Factored form helps us to solve the equation.
16. Solve the following using factoring.
a. x 2  4 x  12 = 0 .
b. x 2  7 x  10 = 0
Show work. x = _______ and x = ________
Show work. x = ______ and x = _______
17. Solve the following quadratics by factoring. First, rewrite the equations in factored form and then
give the solutions to the equations. Remember that the equation must be set equal to zero before
factoring.
a. x 2  4 x  21  0
__________________ b. x 2  5 x  14  0 ____________________
c. x 2  5 x  6  0
__________________ d. x 2  2 x  27  3 ____________________
e. x 2  9 x  36  0
__________________ f. x 2  11x  28  2 ___________________
g. x 2  7 x  60  0
__________________ h. x 2  6 x  9  0
____________________
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Dr. Antonio R. Quesada, Director Project AMP
i. x 2  64  0
__________________ j. 3 x 2  27  0
____________________
Now, we need to be able to write the equation from looking at the graph. We have been finding zeroes on
our graphs and we have seen the connection between the solution to an equation and the graph.
18. Give the equation in factored form for each of the following quadratic graphs.
a. ______________________________
b. ______________________________
c. ______________________________
d. ______________________________
Application Problems
19. The area of a rectangle is given by A  x 2  18 x  72 .
a. Use factoring to find an expression for the dimensions of the rectangle.
b. If the area of the rectangle is 7 square feet, what are the possible values of x?
c. What are the dimensions of the rectangle?
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Dr. Antonio R. Quesada, Director Project AMP
20. Recall the area of a circle is given by A  r 2 , where r is the radius of the circle.
a. If a particular circle is given by A   ( x 2  20 x  100) , find an expression for the radius of the
circle.
b. If the area of the circle is 16 square feet, what is the value of x?
21. The product of two consecutive odd integers is 1 less than four times their sum. Find the two
integers. Hint: There will be two sets of solutions.
22. The hypotenuse of a right triangle is 6 more than the shorter leg. The longer leg is three more than
the shorter leg. Find the length of the shorter leg.
Hint: Draw a right triangle and apply the Pythagorean Theorem.
23. List all definitions, properties, and methods that you learned or reviewed during this investigation.
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Dr. Antonio R. Quesada, Director Project AMP
Extension #1
Sketch a graph of a quadratic function that would not have any real number solutions and explain why
there would not be any real number solutions.
Extension #2
Two cars leave an intersection. One car travels north and the other car travels east. When the car
traveling north had gone 24 miles, the distance between the cars was four miles more than three times the
distance traveled by the car heading east. Find the distance between the cars at that time.