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| 10 Theory slides |
| 8 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Consider the numbers a=2, b=12, and c=10. Find two numbers p and q such that the following conditions are satisfied.
Magdalena's class is planning a camping trip. Magdalena volunteered to research tents to purchase. When she asked her teacher what size they would buy, he instead told her that the base area of the tent should be 3x2+12x+9 square feet and its width w is x+3 feet.
To choose a suitable tent, Magdalena needs to know the dimensions of the tent's base.
Factors of 3 | Sum of the Factors |
---|---|
-1 and -3 | -1+(-3)=-4 × |
1 and 3 | 1+3=4 ✓ |
LHS/(x+3)=RHS/(x+3)
Cancel out common factors
Simplify quotient
Rearrange equation
Distribute 3
Substitution | Calculation | |
---|---|---|
w=x+3 | w=1+3 | w=4 feet |
ℓ=3x+3 | ℓ=3⋅1+3 | ℓ=6 feet |
The class will buy tents that are 4 feet wide and 6 feet long.
seeits factors. Consider the following expression.
It is known that a=4 and c=3, so ac=12>0. Therefore, the factors must have the same sign. Also, b=13. Since the sum of the factors is positive and they must have the same sign, both factors must be positive. All positive factor pairs of 12 can now be listed and their sums checked.
Factors of ac | Sum of Factors |
---|---|
1 and 12 | 1+12=13 ✓ |
2 and 6 | 2+6=8 × |
3 and 4 | 3+4=7 × |
In this case, the correct factor pair is 1 and 12. The following table sums up how to determine the signs of the factors based on the values of ac and b.
ac | b | Factors |
---|---|---|
Positive | Positive | Both positive |
Positive | Negative | Both negative |
Negative | Positive | One positive and one negative. The absolute value of the positive factor is greater. |
Negative | Negative | One positive and one negative. The absolute value of the negative factor is greater. |
Such analysis makes the list of possible factor pairs shorter.
Heichi and Maya are reading a pamphlet about local campsites. They volunteered to find a suitable campground for their class field trip. Their math teacher, who goes camping often, gives them a challenge to help them decide where to go. If they answer his questions about the pamphlet correctly, he will tell them about his favorite local campsite.
Their teacher tells them that the area of the pamphlet is 3x2−5x+2 square inches and that it has a width of x−1 inches.
Now, all the negative factor pairs of 6 will be listed and checked whether their sum is -5.
Factors of ac=6 | Sum of Factors |
---|---|
-1 and -6 | -1+(-6)=-7 |
-2 and -3 | -2+(-3)=-5 |
Commutative Property of Addition
Factor out 3x
Rewrite 2 as 2⋅1
Factor out -2
Factor out (x−1)
Start by writing a quadratic expression for the area of the tarp. Then, factor the obtained expression.
An expression for the area of the tarp can be written by using the expressions for its length and width. Let w be the width of the tarp. Since the length is 1 inch less than twice the width, it can be expressed as 2w−1.
Now, the factor pairs of -56 with opposite signs and where the absolute value of the negative factor is greater than the absolute value of the positive number will be listed. Also, their sum will be calculated to see if it is -1.
Factors of ac=-56 | Sum of Factors |
---|---|
-56 and 1 | -56+1=-55 |
-28 and 2 | -28+2=-26 |
-14 and 4 | -14+4=-10 |
-8 and 7 | -8+7=-1 |
Factor out 2w
Split into factors
Factor out 7
Factor out (w−4)
Use the Zero Product Property
(I): LHS+4=RHS+4
(II): LHS−7=RHS−7
(II): LHS/2=RHS/2
The value of h is zero when the ball hits the ground.
Now, the factor pairs of -30 with opposite signs and where the absolute value of the positive factor is greater than the absolute value of the negative factor will be listed. Also, their sum will be calculated to see whether it is 13.
Factors of ac=-30 | Sum of Factors |
---|---|
-1 and 30 | 29 |
-2 and 15 | 13 |
-3 and 10 | 7 |
-5 and 6 | 1 |
Rearrange equation
Factor out -5t
Rewrite 6 as 2⋅3
Factor out -2
a+(-b)=a−b
Factor out (t−3)
Use the Zero Product Property
(I): LHS+2=RHS+2
(I): LHS/(-5)=RHS/(-5)
(I): Put minus sign in front of fraction
(I): Calculate quotient
(II): LHS+3=RHS+3
There is a jogging track around the campground with the same width on every side. It is known that the campground has a rectangular shape with a width of 30 meters and a length of 50 meters.
LHS−2204=RHS−2204
Rearrange equation
Split into factors
Factor out 4
Factors of -176 | Sum of Factors |
---|---|
-1 and 176 | -1+176=175 |
-2 and 88 | -2+88=86 |
-4 and 44 | -4+44=40 |
-8 and 22 | -8+22=14 |
-11 and 16 | -11+16=5 |
-16 and 11 | -16+11=-5 |
-22 and 8 | -22+8=-14 |
-44 and 4 | -44+4=-40 |
-88 and 2 | -88+2=-86 |
-176 and 1 | -176+1=-175 |
LHS/4=RHS/4
Use the Zero Product Property
(I): LHS+4=RHS+4
(II): LHS−44=RHS−44
Substitute the given numbers a, b, and c into the equations of the conditions.
Factors of 6 | Sum of Factors |
---|---|
1 and 6 | 7 |
2 and 3 | 5 |
A rectangular trampoline with a width of 8 feet and a length of 12 feet is bordered by soft cushions. The width of the square ring made up of the cushions is the same on every side. The area of the surface of the trampoline is equal to the total area of the cushions.
We will start by finding the area of the trampoline. Since it has a rectangular shape, we can find its area by multiplying its length l by its width w.
The trampoline has an area of 92 square feet. We are told that the surface area A_t of the trampoline is equal to the total area A_c of the square ring of cushions around it. This implies that the total area of the trampoline and cushions is twice the area of the trampoline alone.
Now, let's call the cushion width x. This allows us to write an equation representing the total area of the trampoline and cushions. A_(total)= l w ⇕ 192=( 12+x+x)( 8+x+x) Let's simplify this equation and write it in the form ax^2+bx+c=0.
Great! Notice that the simplified quadratic equation has a leading coefficient of 1. We can therefore factor it as (x+ p)(x+ q), where p and q are factor pairs of -24 whose sum is 10.
Factors of -24 | Sum of the Factors |
---|---|
1 and -24 | 1+(-24)=- 23 * |
2 and -12 | 2+(-12)=-10 * |
3 and -8 | 3+(-8)=- 5 * |
4 and -6 | 4+(-6)=- 2 * |
6 and -4 | 6+(-4)=2 * |
8 and -3 | 8+(-3)=5 * |
12 and -2 | 12+(-2)=10 ✓ |
24 and -1 | 24+(-1)=23 * |
According to table, we can substitute p= 12 and q= -2 into our factored equation. Let's factor our equation. x^2+10x-24=0 ⇕ (x+ 12)(x+( -2))=0 Finally, we will apply the Zero Product Property to solve the factored equation for x, the width of the cushions.
Since a negative width does not make sense, we should use the positive solution. The width of the cushion ring is 2 feet.