Algebra 1 Summer Enrichment 2013
C. Joyce
Unit 8: Factoring and Solving Quadratics
Topic 1: Factoring out a GCF
Notes: Read and markup the text. Answer the questions as you go.
Greatest Common Factor (GCF)
The GCF for a polynomial is the largest monomial that divides (is a factor of)
each term of the polynomial.
Example 1: Find the GCF of the list of monomials:
We need to figure out what the largest monomial that we can divide out of
each of these terms would be. Note how they all have an x, so it look like x
will be involved.
our GCF is:
Note that if all terms have the same variable, the GCF for the variable
part is that variable raised to the lowest exponent that is listed.
Example 2: Find the GCF of the list of monomials:
We need to figure out what the largest monomial that we can divide out of
each of these terms would be. What do you think?
So our numerical GCF is:
Now onto the variable part. It looks like each term has an x and a y. In both
cases the lowest exponent is 1.
So the GCF of our variable part is:
Putting this together we have a GCF of:
Factoring out the GCF
Step 1: Identify the GCF of the polynomial.
Step 2: Divide the GCF out of every term of the polynomial.
Example 3: Factor out the GCF:
Step 1: Identify the GCF of the polynomial.
The largest monomial that we can factor out of each term is:
Step 2: Divide the GCF out of every term of the polynomial.
*Divide 2x out of every term of the poly.
Be careful. If a term of the polynomial is exactly the same as the GCF,
when you divide it by the GCF you are left with 1, NOT 0. Don’t think,
'oh I have nothing left', there is actually a 1. As shown above when we divide
2x by 2x we get 1, so we need a 1 as the third term inside of the ( ).
Note that if we multiply our answer out, we should get the original
polynomial. In this case, it does check out. Factoring gives you another way
to write the expression so it will be equivalent to the original problem.
Practice: Factor out the GCF of each polynomial:
1. 4𝑦 3 − 12𝑦 + 8𝑦 2
2. 2𝑥𝑦 4 − 6𝑥 2 𝑦 3 + 8𝑥 3 𝑦 2
3. Factor out the GCF: (16𝑥 2 𝑦 3 + 6𝑥𝑦 7 )
4. Fill in the ovals with the GCFs of the rows and columns. The first column has
been done for you!
2𝑥
6𝑥 2
15𝑥
2𝑥
5
DIAMOND PUZZLES
To solve the diamond puzzles, you multiply the left and right number to get the
top number. You added the left and right number to get the bottom number.
For example: 5 ∙ 2 = 10, 5 + 2 = 7
The puzzles below follow the same pattern. Try to fill in the left and right
numbers!
Medium
Topic 2: Factoring Quadratics
Basic Factoring Quadratics when the Leading Coefficient is 1
Factoring a quadratic is just like playing a game. You’re looking for 2 numbers that are factors of
the constant that add to give you the middle term.
Example:
x 2  14 x  24
Factors of 24
Add to
1 x 24
2 x 12
3x4
25
14
12
The correct numbers are
+2 and +12, we express
these answers as:
Answer:
 x  2 x  12
Practice:
1. 𝑥 2 + 9𝑥 + 18
2. 𝑥 2 + 2𝑥 − 8
Part 2: CHALLENGE!
The area model’s sum is 2𝑥 2 + 7𝑥 + 3. We can fill in some of the boxes of our area model with
this information, but we need to figure out the missing numbers. Fill in the missing numbers
on the area model.
2𝑥
𝑥
2𝑥 2
3
Make a diamond puzzle using the numbers in the ovals.
1. Multiply the numbers in the ovals and put them in the top box.
2. Add the numbers in the ovals and put them in the bottom box.
What do you notice about the numbers in your diamond puzzle and
the numbers in your original expression: 2𝑥 2 + 7𝑥 + 3?
Factoring all quadratics!!
Example
Factor: 2x2 – 7x - 4
Steps
1st: Ask yourself: “Is there a GCF?”
If the answer is yes, just factor out
the GCF.
BUT, if the answer is NO…
a= 2
b = -7
c = -4
2nd: Identify a, b, and c.
𝑎∙𝑐
-8
-8
1
3rd: Complete a diamond puzzle.
-7
𝑏
x
2x
1
2x2
1x
( 2x +
-4
-8x
-4
1 )( x – 4 )
4th: Write your answers from the
diamond puzzle in the area model.
5th: Find the GCF of each pair in the
box.
6th: Write each side as a binomial in
the parenthesis.
Example
Example
Factor: 3x + 8x + 5
Factor: x2 – 7x + 12
2
a=
b=
c=
(
)(
)
*Challenge*: Factor 6x2 – x – 15.
a=
b=
c=
(
)(
)
My Factoring Resource
This page is YOUR resource for factoring. Make it meaningful! Use words and
phrases that you will remember. Try and write hints to yourself that will help you!
Factor 𝟔𝒙𝟐 + 𝟏𝟕𝒙 + 𝟓.
Explain each step of your process in
your own words. If there is something
you want to stand out please write it in
another color.
More Factoring Practice – Factor FIVE of the following expressions. (Label your
work and use extra paper if you need to! Just staple it to the back.)
1. 3x² + 7x + 2
2. x² + 2x – 6
3. x² - 21x – 72
4. 7x² - 9x + 2
5. 3x² - 2x – 5
6. 33x² - x - 14
7. 21x² + 22x – 24
8. 4x² + 8x + 4
9. 7x² + 5x - 56
Challenge: x² - 49
Challenge
There’s More to Factoring…..
Examine the image below and in the box on the right jot down some things that you learned from
looking at the image.
Part 1: Difference of Squares
A.) Steps to Factoring Difference of Squares
1. Are there only 2 terms?
2. Is the first term a perfect square?
3. Is the last term a perfect square?
4. Is there subtraction (difference) in the problem?
If all of these are true, you can factor using this
method!!!
B.) Factor the following Difference of Squares:
𝟏. 𝒙𝟐 − 𝟐𝟓
2. 𝟗𝒙𝟐 − 𝟐𝟓𝒚𝟐
3. 𝟒𝒙𝟐 + 𝒚𝟐
Part 2: Perfect Square Trinomials
A.) Perfect Square Trinomials can be factored just
like other trinomials (guess and check), but if
you recognize the perfect squares pattern,
follow the formula!
B.) Factor the following Perfect Square Trinomials.
4. 𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐𝟓
5. 𝟒𝒎𝟐 − 𝟏𝟐𝒎 + 𝟗
6. 𝟗𝒙𝟐 + 𝟔𝒙 + 𝟏
Part 3: More Challenging Factoring Problems – factor completely
7. 𝒚𝟑 − 𝟑𝟔𝒚
8. −𝟐𝒙𝟐 + 𝒙 + 𝟏𝟎
9. 𝟑𝒙𝟒 − 𝟕𝟓
**10. Find the value of c that makes cn2 + 72ny + 81y2 a perfect square trinomial.
Topic 3: Solving Quadratics by Factoring
We need a way to solve quadratic equations! We can do this using factoring.
BRAINSTORM:
1.
List three pairs of numbers that multiply to 0.
2.
What do you notice about the pairs of numbers?
We are going to apply this idea to solving quadratic equations!
Notes: Read and markup the steps below and take your notes.
SOLVING BY FACTORING PROCESS:
STEP 1:
Set the quadratic equation = 0
STEP 2:
Factor the quadratic
STEP 3:
Set each factor = 0
STEP 4:
Solve each equation
STEP 5:
If it is a word problem, make
sure you check to see which
answer(s) make sense.
2𝑥 2 − 7𝑥 − 4
2𝑥 2 − 7𝑥 − 4 = 0
(2𝑥 + 1)(𝑥 − 4) = 0
2𝑥 + 1 = 0 𝑎𝑛𝑑 𝑥 − 4 = 0
2𝑥 + 1 = 0 𝑎𝑛𝑑 𝑥 − 4 = 0
-1 -1
+4 +4
2x = 1
x= ½
and x=4
Not Applicable
Your notes
Instructions: Choose a level (either medium or spicy) and solve the equations in
your section by factoring.
(3𝑥 + 6)(7𝑥 − 14) = 0
MEDIUM:
2𝑥 2 + 9𝑥 = 0
SPICY:
𝑥 2 + 3𝑥 = 10
𝑥(3𝑥 + 9)(2𝑥 − 4) = 0
𝑥 3 − 3𝑥 2 + 2𝑥 = 0
3𝑥 2 − 2𝑥 − 1 = 0
(𝑥 − 1)(𝑥 + 2) = 28
6𝑥 2 + 𝑥 = 2
4𝑥 3 + 6𝑥 2 − 4𝑥 = 0
3𝑥 2 = 7𝑥 − 2
Review for Factoring
Section 1: Multiple Choice
2 2
3
4
1. Which of the following is the GCF of the polynomial? 4a b  6ab  2ab
2 4
B.) 6a b
A.) 2ab
x 2  16
2. Which of the following are the correct factors of
2
A.) ( x  4)
2
D.) 2ab
2
C.) ab
B.) ( x  4)( x  4)
2
C.) 2( x  4) D.) ( x  8)
3. Which of the following is the sum of both solutions of the equation?
x 2  3 x  54  0
A.) -3
B.) 3
C.) -15
D.) 15
Section 2: Solving quadratics using Factoring
Find the factors and solutions of the following quadratic equations.
2
4. x  7 x  12  0
Factors: (
Solutions:
2
5. 3 x  4 x  7  0
)(
x=
)
x=
Factors: (
Solutions:
)(
x=
)
x=
2
6. 6 x  2 x  8  0
Factors: (
Solutions:
3
7. a  36a
)(
x=
)
Factors: (
x=
Solutions:
)(
x=
)
x=
___________________________________________________________________
8. Examine the equation and explain if it is true or false and how you know.
m2  6mn  8n2  (m  4n)(m  2n)
Topic 4: Solving Quadratics by Factoring and Quadratic Formula
Part 1: Factoring to Solve
If you can factor a quadratic you can use the factors to find the zeroes of a quadratic, which in turn tell
us where the x-intercepts are on the graph.
Example:
3x 2  5 x  2  0
factors:
 3x  1  x  2 
0
3x  1  0 x  2  0
x
1
3
x  2
1
3


This means the quadratic equation above has 2 x-intercepts of  , 0  and  2,0 
Part 2: Quadratic Formula to Solve
The Quadratic Formula can also give you the x-intercepts. The first thing we must do is to put the
quadratic into standard form, which looks like: y  ax  bx  c
2
Then we use the coefficients to complete the formula:
x
b  b2  4ac
2a
This formula is VERY IMPORTANT!!! Take time to remember what it is and where each letter comes
from.
Example:
Since x  2 or 3 , we know that the graph will
have x-intercepts at (2,0) and (3,0). We can
now write the factored form of the original
equation by solving for zero.
f ( x)  x 2  5 x  6
x
  5  
 5
2 1
2
 4 1 6 
5  25  24 5  1 5  1


2
2
2
5 1 6
5 1 4
x
  3, and x 
 2
2
2
2
2
x  3 and 2
x
Ex. x  3  x  3  0 , so x  3 is a factor.
x  2  x  2  0 , so x  2 is also a
factor
This means that the factored form of the
equation is y  x  3x  2
Independent Practice
1) Factor and Solve: h( x)  2 x 2  8 x  10
2) Factor and Solve k ( x)  4 x 2  4 x  3
Use the quadratic formula to find the x-intercepts and write the equation in factored form.
3) r ( x)  4 x 2  13x  3
4) s( x)   x 2  11x  12
Solving Quadratic Equations Using Factoring and Quadratic Formula
Section 1: Solve using factoring.
1.
x2  6x  8  0
2.
x 2  10 x  21  0
Section 2: Solve using quadratic formula.
3.
5x2  6 x  8  0
4. 12 x
5.
2 x 2  3x  5  0
6.
x 2  x  3.75  0
7.
x2  4x  4  0
8.
4 x2  8x  7  4 
2
 41x  24  0
Topic 5: Simplifying Roots
Notes: Read and markup the diagram below and take your own notes.
Simplifying Roots
Your notes
Topic 6: More Quadratic Formula
Answer the questions below:
Station 1
1. At this station you will have 6 index cards. As
a group flip over the 6 index cards and
match each quadratic equation with the
appropriate quadratic formula. Write the
matches below.
Station 2
1. At this station you will have a dice. Roll
the dice once and write the number in
the first box. Roll it again and write the
number in the second box.
2. You can find the number and type of
solutions to a quadratic equation by
finding the discriminant. The
discriminant is the value under the
square root in the quadratic formula
(𝑏 2 − 4𝑎𝑐). Find the discriminant of the
quadratic equation you wrote in number
1.
2. Choose ONE of the quadratic equations and
solve it below.
3. Based on your discriminant, what type of
solution(s) does the quadratic function
you created have? Explain how you
know.
Station 3
1. At this station you find a word problem. The
equation to model the word problem is:
Solve the quadratic function using the
quadratic formula.
Station 4
1. At this station you will get a card with 4
scenarios, you need to determine
whether there are two, one or no REAL
solutions of each equation. Write down
each scenario first!
Scenario A:
How many real solutions?
Scenario B:
2. Which solution gives the amount of time it
takes for the baseball to hit the ground?
How many real solutions?
Scenario C:
3. Why will only one of your solutions equal
the time it takes for the baseball to hit the
ground?
How many real solutions?
Scenario D:
How many real solutions?
Summary – use what you learned today in these stations to answer the questions that follow.
9𝑥 2 − 30𝑥 + 25 = 0
1. Find the discriminant.
2. Based on the discriminant describe how many real solutions the quadratic equation has.
Explain!
3. Solve using quadratic formula.
Quadratic Formula with Non Perfect Roots:
Sometimes when you complete a problem the square root does not have a perfect square such as
16  4 or
30  ??? When this happens, you can stop the problem once you simplify the square
root as much as you can.
In this example when you simplify inside the
Example:
2 x 2  6 x  9  0
x
x
x
6 
 6
2
 4  2  9 
2  2 
6  36  72
4
6  108
4
square root, you end up with 108 which does
not have a perfect square root. When you reduce
the radical you end up with 6 3 then you can
make the two branches and get
6  6 3
6  6 3
and
which when you reduce
4
4
the whole numbers you get
3  3 3
3  3 3
and
2
2
Solve each quadratic equation using the quadratic formula:
5. x 2  11x  16  0
6. 3 x 2  x  1  0
7. 3 x 2  x  5  0
8. x 2  3 x  8  0
9. 5 x 2  8 x  6  0
10. 3 x 2  8 x  12  0
Topic 7: Applications of Quadratic Equations
Notes: Read and markup the example and take your own notes
Example:
1. The equation 𝒅 = 𝟎. 𝟎𝟓𝒗𝟐 + 𝟏. 𝟏𝒗 models the distance
in feet it takes a car traveling at a speed of v miles per hour to
come to a complete stop. If Hannah’s car stopped after 250 feet
on a highway with a speed limit of 65 miles per hour, was she
speeding? Explain your reasoning.
a.) List the 2 variables and define what they mean in context.
v = speed in miles per hour
d = distance to come to a complete stop in feet
b.) Which one is the x-variable and which one is the y-variable?
Explain.
The x variable is v or speed because it is what comes first and the
y variable is d or distance to stop because it happens in response.
c.) What is the main idea of this question?
Is Hannah speeding? This means is her speed (v) greater than the
speed limit of 65 miles per hour. So we need to find if v > 65. If it is,
she was speeding; if it is not than she was not speeding.
Your Notes
d.) How can you use the equation, quadratic formula, and the
numbers provided to answer the question? Explain and show
your work.
d = 250 so plug it in for d in the equation then solve using
quadratic formula.
250 = 0.05𝑣 2 + 1.1𝑣
0.05𝑣 2 + 1.1𝑣 − 250 = 0
𝑥=
−1.1 ± √1.12 − 4(0.05)(−250)
2(0.05)
𝑥=
−1.1 ± 7.2
.1
𝑥 = 61 𝑎𝑛𝑑 𝑥 = −83
e. Answer the question.
Since -83 miles per hour makes no sense I will throw out
that answer in favor of the x = 61 miles per hour. This means
Hannah was not speeding because her speed was less than
the speed limit of 65 mph.
Your Turn:
2. At liftoff, the space shuttle Discovery has a constant acceleration of 16.4 feet per second squared
and an initial velocity of 1,341 feet per second due to the rotation of the Earth. The distance Discovery
has traveled t seconds after liftoff is given by the equation 𝒅(𝒕) = 𝟏𝟑𝟒𝟏𝒕 + 𝟖. 𝟐𝒕𝟐 . How long after
liftoff has Discovery traveled 40,000 feet?
a) List the 2 variables and define what they mean in context.
b) Which one is the x-variable and which one is the y-variable? Explain.
c) What is the main idea of this question?
d) How can you use the equation, quadratic formula, and the numbers provided to answer the
question? Explain and show your work.
e) Answer the question.