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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 |
Paulina wants to buy a mirror to tie in the decorations of her living room. The mirror's area in square feet is represented by 4x2+5x−6 and its width is x+2 feet.
Note that the shape of the mirror is a rectangle. This means we can substitute the given expressions for the area and the width into the formula for the area of a rectangle to write an equation. A= l w ⇓ 4x^2+5x-6= l( x+2) We can rewrite the trinomial on the left-hand side of the equation so that the width is one of the factors and the other will be the expression for the length. Then we will factor by grouping. Let's determine the values of a, b, and c of the trinomial. 4x^2+5x-6 ⇕ 4x^2+ 5x+( - 6) Here, a= 4, b= 5, and c= -6. With this in mind, we can make the following conclusions.
Using this information, let's now list the factor pairs of -24 and look for the sum of 5.
Factors of ac=-24 | Sum of Factors |
---|---|
24 and -1 | 24+(- 1)=23 |
12 and -2 | 12+(- 2)=10 |
8 and -3 | 8+(- 3)=5 |
6 and -4 | 6+(- 4)=2 |
Now that we know which factor pair to use, we can rewrite bx as the sum of two terms by using the factors 8 and -3. 4x^2 + 5x-6 ⇕ 4x^2 - 3x + 8x-6 Next, we will factor the new expression by grouping its terms.
Now we have factored the given trinomial. Note that one of the factors, x+2, is the given expression for the width of the mirror. This means that the expression 4x-3 represents the length! Using this information, we can finish writing the equation for the area of the mirror. 4x^2+5x-6= l( x+2) ⇕ 4x^2+5x-6=( 4x-3)( x+2)
One way to find the area of the mirror when x= 2 feet is to substitute 2 into the given expression for the area 4x^2+5x-6 and then simplify. Let's do it.
The area of the mirror is 20 square feet when x= 2 feet. Notice that we can also find the same area by substituting x= 2 into the product of the factors x+2 and 4x-3.
The length ℓ of a signboard is two feet less than three times its width w. The area of the signboard is 21 square feet.
We are told that the length l of the signboard is two feet less than three times its width w. Using this information, we can write an expression for l. l= 3w-2 The shape of the signboard is a rectangle. This means its area is given by the product of its length l and its width w. A= l w Additionally, we are given that the area of the signboard is 21. Therefore, we can substitute 21 for A and 3w-2 for l in the formula for the area to get an equation for the area of the signboard. A= l w Substitute 21=( 3w-2) w We will now rewrite the this equation in the form aw^2+bw+c=0. This will help us find the value of w.
The trinomial on the left-hand side of the equation can now be factored. To do so, let's first identify its coefficients. 3w^2-2w-21=0 ⇕ 3w^2+( -2)w+( - 21) =0 Here, a= 3, b= -2, and c= -21. With this in mind, we can make the following conclusions.
We will list the factor pairs of -63 and look a pair for a sum of -2.
Factors of ac=-63 | Sum of Factors |
---|---|
-63 and 1 | (- 63)+1=-62 |
-21 and 3 | (- 21)+3=-18 |
-9 and 7 | (- 9)+7=-2 |
Now that we know which factor pair to use, we can rewrite bw as the sum of two terms by using the factors -9 and 7. 3w^2 - 2w-21=0 ⇕ 3w^2 - 9w + 7w-21=0 Next, we will factor the expression by grouping its terms.
Notice that the product of these two factor expressions is zero. This means that we can use the Zero Product Property to identify the solutions to the equation.
We will use the positive solution since a negative width does not make sense. Therefore, the width of the signboard is 3 feet.
Consider the equation for l that we wrote in Part A. l=3w-2 We also found that the width of the signboard is 3 feet. We will substitute this value for w in the equation and simplify to find the value of l.
The length of the signboard is 7 feet.
Let's start this exercise by assuming that the water is at ground level h=0. Then we can factor the given equation to solve for the time it takes to hit the water. h=-10t^2+5t+15 ⇓ 0 =-10t^2+5t+15 The terms of the quadratic equation have as a common factor, -5. Therefore, we will rewrite this equation by factoring that out.
The obtained trinomial on the left-hand side of the equation needs to be factored to identify the solutions of the equation. To do so, let's first identify the coefficients of its terms. 2t^2-t-3=0 ⇕ 2t^2+( -1)t+( - 3) =0 Here, a= 2, b= -1, and c= -3. With this in mind, we can make the following conclusions.
Using this information, let's list the factor pairs of -6 and look for the pair with a sum of -1.
Factors of ac=-6 | Sum of Factors |
---|---|
-6 and 1 | (- 6)+1=-5 |
-3 and 2 | (- 3)+2=-1 |
Now that we know which factor pair to use, we can rewrite bt as the sum of two terms by using the factors -3 and 2. 2t^2 - 1t-3=0 ⇕ 2t^2 + 2t - 3t -3=0 Next, we will factor the expression by grouping its terms.
Notice that the product of the factor expressions is zero. This means that we can use the Zero Product Property to set each factor equal to 0 and solve for t.
The negative solution is not relevant in this case because time cannot be negative. Therefore, the diver will reach the water 1.5 seconds after the they start the dive.