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Activity 2.2.3 Solve Equations by Factoring
When you multiply two or more factors and the result is zero, then at least one of the factors has
to be ____.
Example: given that (x + 5)(x - 2)= 0, then by the special property of zero either x + 5 = 0 or
x-2=0
So we solve:
x – 2= 0
x + 5=0
x = ___
or
x =____
Check to see if (x + 5)(x - 2)= 0 is a true statement if -5 is substituted in for x:
Check to see if (x + 5)(x - 2)= 0 is a true statement if 2 is substituted in for x:
The two solutions to the equation (x + 5)(x - 2)= 0 are ____ and _____.
To solve the quadratic equation in one variable by factoring: If needed obtain an equivalent
equation with one side 0, factor the nonzero side into linear factors, set each linear factor = 0,
solve.
A. SOLVE:
1.
 x  4 x  7  0
3. x 2  12 x  11  0
Activity 2.2.3
2.
 2 p 13 p  5  0
4. z 2  7 z  18
Connecticut Core Algebra 2 Curriculum Version 3.0
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5. 6k 2  13k  5
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6.
 m  2 m  1  12
Solve each equation by factoring, if possible. If not factorable over the integers, write “trinomial
is prime”.
7. 𝑑 2 + 6𝑑 + 9 = 0
8. 3(𝑥 2 + 7𝑥 − 1)+5=2(x+1)
9. c 2  5c  7
10. 5 x 2  13x  6
Activity 2.2.3
Connecticut Core Algebra 2 Curriculum Version 3.0
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11. 9𝑥 2 − 16 = 0
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12. 3x 2  4 x  8
13. Here are two quadratic equations that cannot be solved by factoring over the integers. One
equation has no real number solutions. The other equation has two irrational, real solutions.
Which one is which? Justify your answer.
a. 𝑥 2 + 𝑥 − 1 = 0
b. 𝑥 2 + 𝑥 + 1 = 0
B. Now, let’s work backwards.
If you are given the answers to a quadratic equation, how can you find the equation?
Suppose the solutions to a quadratic equation are 2 and -5. What does a quadratic equation look
like just before you find the solutions?
______________________________________________________________________
How does that become a quadratic equation? __________________________________
Is there another equation that has the same solutions?______________________
Write a quadratic equation first in factored form, then multiply out to obtain a quadratic with
integral coefficients in the form ax 2  bx  c  0 with the given solutions.
14. x  4, 7
Activity 2.2.3
15. x 
2
,6
3
Connecticut Core Algebra 2 Curriculum Version 3.0
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16. x  0, 
Date:
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17. 𝑥 = −2, 𝑎𝑛𝑑 − 2
C. Mixed Practice: solve the following linear and quadratic equations:
18. 2m 2  14m
20.
22.
 f  5 f  1   f  3 f  7 
2𝑥 2 + 3𝑥 − 11 = 2(𝑥 − 1)(𝑥 + 1)
Activity 2.2.3
19. 2r+64=16r
21. 𝑟 2 − 64 = 0
23. r 2  64  16r
Connecticut Core Algebra 2 Curriculum Version 3.0
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24. If the two solutions of a quadratic equation are 5 and -7, what is an equation with those
solutions?
D. Summary:
1. How can you tell if an equation in one variable is linear or quadratic? _______________
a. After simplifying, a linear equation has a degree of ______ and will have _______ solution(s).
b. After simplifying, a quadratic equation has a degree of _______ and will have at
most ______ solution(s).
2. How do you solve a linear equation in one variable?
3. How do you solve a quadratic equation in one variable using the factoring method?
Activity 2.2.3
Connecticut Core Algebra 2 Curriculum Version 3.0