3.3.4 Practice: Modeling: Factoring ax2 + bx + c
Algebra I Honors
Points Possible:30
Practice
Name: Bradphil Guillaume
Date:
Your Assignment: Patio Project
You have a small patio and want to improve it with a project. Circle the project you picked:
Fish Pond
Planter Box
Fire Pit
Making Sense of the Problem
Make Sense of the Problem (3 points: 1 point for each answer)
Patio area: 24 square feet
Shapes: both the patio and the fire pit are
rectangular
What do you know?
Length of the fire pit is twice the width
1-foot border around the fire pit
2x^2+6x+4 is the total area of the patio with x
being the width of the fire pit
What do you want to find out?
I want to find out x which represents the width
and the length which can be represented by 2x
I expect the answer to be in feet and represent
Wh.xat kind of answer do you expect? realistic dimensions of the fire pit.
For example, usually a fire pit measures
approximately 2 feet by 1 foot.
Writing an Equation
1. Set your polynomial equal to the total area of your patio to create the equation. (1 point)
2x2+6x+4= 24 square feet
Getting the Right Side of the Equation Equal to 0
2. Use inverse operations to make the right side of the equation equal to 0. (2 points)
2x2+6x-20=0
Factoring the GCF out of the Polynomial
3. Factor the GCF out of the trinomial on the left side of the equation. (2 points: 1 for the GCF, 1
for the trinomial)
2(x2+3x-10)=0
Factoring the Polynomial Completely
4. Factor the polynomial completely. (4 points: 2 points for each factor)
2(x-2)(x+5)=0
5. If a product is equal to zero, we know at least one of the factors must be zero. And the
constant factor cannot be zero. So set each binomial factor equal to 0 and solve for x, the width
of your project. (2 points: 1 point for each factor)
2(x-2)(x+5)=0
x-2=0 or x+5=0
X=2 or x=-5
6. Check your work. Substitute the two values of x into the equation from question 2 to confirm
that it equals zero. (2 points: 1 point for each value of x)
2*(2-2)*(2+5)=0
2-2=0
2+05=7
2*0*7=0
0=0 the solution is correct
2*(-5-2)*(-5+5)=0
-5-2=-7
-5+5=0
2*-7*0=0
0=0 the solution is also correct
7. What are the dimensions of your project? Remember that the width of your project is
represented by x. (2 points: 1 point for each dimension)
Width = 2 feet (the other solution is negative which is -5, so it cannot be the width)
Length = 2*2= 4 feet
8. Draw a new picture of your patio that shows the dimensions of your project and the border. (2
points) Check to confirm that the total area equals the area of your patio.
9. Your friend Mel says he has an easier way to solve this problem. Here are the first steps in his
method.

First, isolate all of the x-terms on the left side. This leaves a constant on the right that isn't
equal to 0.

Factor the GCF out of the terms on the left side.

If there's a whole-number factor on the left side, divide both sides by that number. Notice
that after you do this, one factor on the left is just x.
Apply Mel's steps to your equation in question 1. (5 points)
2x^2 + 6x + 4 – 24
2x^2 + 6x – 20 = 0
2(x^2 + 3x – 10) = 0
2(x – 2)(x+ 5) = 0
x = 2, –5
10. Now, Mel says, you can find all the whole-number factor pairs for the constant number on
the right side. Substitute the smaller factor in each pair for x. If the resulting equation is true,
you've found a solution.
Mel gives you a simple example:
Solve x(x + 1) = 6
The factor pairs for 6 are 1 and 6 and 2 and 3.
Try substituting 1 for x:
1(1 + 1) = 6
1(2) = 6
This is false, so x = 1 is not a solution.
Try substituting 2 for x:
2(2 + 1) = 6
2(3) = 6
This is true, so x = 2 is a solution.
Apply Mel's method to solve for x, the width. Does his method find the correct width? (3 points)
2 (2 + 1) = 2 FALSE
With Mels method the solution was 6 but the real solution is 2.
11. Mel is clever, but you're pretty sure his method isn't safe to use in all situations. Explain why
his method might miss some solutions. (Hint: Compare your answer to question 10 with your
answer to question 5.) (2 points)
With mels method I missed the x solution and this is because in his way you are using factors
and if one pair of factors doesn’t work then you use the other but since 2 is a prime number it
only has one factor pair so if you can’t find x with the factor 1 and 2 then there is no other option
and you can’t find X.
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