2-1. For the entire rectangle below, find the area of each part and then find the area of the whole.
2-2. Find the total area of each rectangle below. Each number inside the rectangle represents the
area of that smaller rectangle, while each number along the side represents the length of that
portion of the side.
2-3. Diagrams like the ones above are referred to as generic rectangles. Generic rectangles allow
you to use an area model to multiply expressions. Draw each of the following generic rectangles
on your paper. Then find the area of each part and write the area of the whole rectangle as
a product and as a sum.
a.
b.
d.
c.
e.
f.
2-4. Multiply the following expressions using either a generic rectangle or the Distributive
Property.
a. 6(2x + y − 5)
b. 3x(6x − 11)
c. −2(15x2 – 3x + 4)
d. (2y2 – 3y + 5)6
2-5. Solve each system of equations below.
a. y = 3x + 1
x + 2y = −5
b. 2x + 3y = 9
x − 2y = 1
c. 3x + y = −10
5x − y = 2
2-6. Finding and using a pattern is an important problem-solving skill you will use in algebra. The
patterns in Diamond Problems will be used later in the course to solve other types of algebraic
problems.
Look for a pattern in the first three diamonds below. For the fourth diamond, explain how you
could find the missing numbers (?) if you know the two numbers (#).
Copy the Diamond Problems below onto your paper. Then use the pattern you discovered to
complete each one.
2-7. Multiply the following expressions using either a generic rectangle or the Distributive
Property.
a. 5(2x2 + x − 3)
b. 10x(4 − 2x)
c. −3(5x2 – 9x + 1)
d. 3(7x2 – 8x + 2)
2-8. Work with your team to solve each of these equations. Use the Distributive Property or draw
generic rectangles to help you rewrite the products. Be sure to record your algebra work for each
step.
2(y − 2) = −6
b. 2(x + 1) + 3 = 3(x − 1)
c. 2(x + 4) = 4(x + 2)
d. 2(3x – 3) = 3(2x − 2)
a.
2-9. Write a system of equations to solve.
Thanh bought 11 pieces of fruit and spent $5.60. If apples cost $0.60 each and pears cost $0.35
each, how many of each kind of fruit did he buy?
A generic rectangle can be used to find products because it helps to organize the different areas that make up the
total rectangle. For example, to multiply (2x + 5)(x + 3), a generic rectangle can be set up and completed as
shown below. Notice that each product in the generic rectangle represents the area of that part of the rectangle.
2-10. Draw each of the following generic rectangles on your paper. Then find the area of each part
and write the area of the whole rectangle as a product and as a sum.
2-11. Multiply and simplify the following expressions using either a generic rectangle or the
Distributive Property.
a. (x + 5)(3x + 2)
b. (2y − 5)(5y + 7)
c. (5w − 2)(3w + 4)
2-12. Solve the following systems algebraically.
a. x + 2y = 17
x−y=2
b. 4x + 5y = 11
2x + 6y = 16
c. 2x + y = −2x + 5
3x + 2y = 2x + 3y
2-13. Find the dimensions of the generic rectangles below. Then write an equivalency statement
(length · width = area) of the area as a product and as a sum.
a.
b.
c.
2-14. Use generic rectangles or the distributive property to multiply the following expressions.
Write each solution both as a sum and as a product.
a.
b.
c.
d.
(2x + 5)(x + 6)
(m − 3)(3m + 5)
(12x + 1)(x − 5)
(3 − 5y)(2 + y)
2-15. Copy and complete each of the Diamond Problems below. The pattern used in the Diamond
Problems is shown beow.
2-16. Solve each equation. Use the Distributive Property or draw generic rectangles to help you
rewrite the products.
a. 3x + 2 = 10 – 4(x – 1)
c. 6(2x – 5) = –(x + 4)
b. 4(x – 1) – 2(3x + 5) = –3x +1
d. (x + 1)(x – 7) = (x – 1)(x + 3)
2-17. Zachary has $718 in his bank account and automatically withdraws (subtracts) $14 every
month to pay for his computer service. Christian has $212 in his bank account and deposits (adds)
$32 each month from his newspaper delivery tips. Assuming they make no other deposits or
withdrawals, when will Zachary and Christian have the same amount of money in their bank
accounts?
2-18. Use generic rectangles or the distributive property to multiply the following expressions.
Write each solution both as a sum and as a product.
a. (x + 5)(x – 5)
e. (3y – 7)2 means (3y – 7)(3y – 7)
b. (m − 6)(m – 6)
f. (2x + 9y)(2x – 9y)
c. (2x + 3)(2x + 3)
g. (z – 1)(z + 1)
d. (3 − 5y)(3 + 5y)
h. (x + 2y)2
2-19. Copy each of the generic rectangles below and fill in the missing dimensions and areas.
Then write the entire area as a product and as a sum.
a.
b.
c.
2-20. Copy these Diamond Problems and use the pattern you discovered earlier, shown below, to
complete each of them.
2-21. Solve each system of equations below.
a. 2x + y = –7y
y = x + 10
b. 3x + 5y = 0
6x – 7y = 17
c. 2x – y = 9
y=x–7
2-22. Solve each equation after first rewriting it in a simpler equivalent form.
a. 3(2x − 1) + 12 = 4x − 3
b. 4x(x − 2) = (2x + 1)(2x − 3)
2-23. Use generic rectangles or the distributive property to multiply the following expressions.
Write each solution both as a sum and as a product.
a. (2x − 1)(2x + 1)
b. (5s − 3t)2
c. (6 − y)(6 + y)
d. (3 + m)2
2-24. Copy and complete each of the Diamond Problems below.
2-25. Write an equation or system of equations to solve this problem.
An adult ticket to the amusement park costs $24.95 and a child’s ticket costs $15.95. A group of 10
people paid $186.50 to enter the park. How many were adults?
2-26. Jackie and Alexa were working on homework together when Jackie said, “I got x = 5 as the
solution, but it looks like you got something different. Which solution is right?”
“I think you made a mistake,” said Alexa. Did Jackie make a mistake? Help Jackie figure out
whether she made a mistake and, if she did, explain her mistake and show her how to solve the
equation correctly. Jackie’s work is shown below.
2-27. Greta is opening a savings account. She starts with $100 and plans to add $50 each week.
Write an equation she can use to calculate the amount of money she will have after any number of
weeks. How much money will she have after 1 year.
2-28. The Distributive Property and common factors can be used to change expressions written as
sums into expressions written as products. For example:
Since 6 is the greatest common factor of both terms, 12x + 18 may be rewritten 6(2x + 3)
Since x is a common factor of every term, x2 + xy + x may be rewritten x(x + y + 1)
Use the greatest common factor to rewrite each sum as a product.
a.
b.
c.
d.
4x + 8
10x + 25y + 5
2x2 − 8x
9x2y + 12x +3xy
e.
f.
g.
h.
20y – 8
4x2 + 6x – 12
10y2 – 20y + 15
8xy + 16y – 12x2
2-29. Multiply each of the following expressions. Show all of your work.
a. (x + 3)(4x + 5)
b. (−2x – 4y)(3x + 4y)
c. (3y − 8)(3y + 8)
d. (x − 4)(3x2 + 5x − 2)
2-30. Copy and complete each of the Diamond Problems below.
2-31. Solve each of the following equations. Be sure to show your work carefully and check your
answers.
a. 2(3x − 4) = 22
c. 2 − (y + 2) = 3y
b. 6(2x − 5) = −(x + 4)
d. 3 + 4(x + 1) = 159
2-32.Herman and Jacquita are each saving money to pay for college. Herman currently has
$18,000 and is working hard to save $1000 per month. Jacquita only has $5,000 but is saving
$2000 per month. In how many months will they have the same amount of savings?
2-33. Solve each system of equations.
a.
b.
c.
d.
2-34. Rewrite each of these products as a sum.
a. 6(2x2 + 5x − 8)
b. (2x − 11)(x + 4)
c. (7x – 4)(7x + 4)
d. (x − 2)2
2-35.Charles and Amy are part of the Environmental Club at school. Charles’ uncle owns a tree
nursery and is willing to donate a 3-foot tall tree that he says will grow 1.5 feet per year. Amy goes
to another nursery in town, but is only able to get tree seeds donated. According to the seed
package, the tree will grow 1.75 feet per year. Charles plants his tree and Amy plants a seed on the
same day. Amy thinks that even though her tree will be much shorter than Charles’ tree for the first
several years, it will eventually be taller because it grows more each year, but she does not know
how many years it will take for her tree to get as tall as Charles’ tree.
Will the trees ever be the same size? If so, how many years will it take?
2-36. For each of the following generic rectangles, find the dimensions (length and width) and
write the area as the product of the dimensions and as a sum.
a.
b.
When an expression is written in product form, it is said to be factored and each of the expressions
being multiplied is called a factor. The factored form of x2 − 15x + 26 is (x − 13)(x − 2),
so (x – 13) and (x – 2) are each factors of the original expression.
2-37. Work with your team to find the sum and the product for the following generic rectangles.
Are there any special strategies you discovered that can help you determine the dimensions of the
rectangle?
a.
b.
c.
2-38. While working on problem 2-37, Casey noticed a pattern with the diagonals of each generic
rectangle. However, just before she shared her pattern with the rest of her team, she was called out
of class! The drawing on her paper looked like the diagram below. Can you figure out what the
two diagonals have in common?
2-39. Use products and sums or generic rectangles to factor the following expressions.
a. x2 − 4x – 12
b. x2 + 9x + 18
c. k2 − 12k + 20
d. x2 − 8x + 16
2-40. Remember that a Diamond Problem is a pattern for which the product of two numbers is
placed on top, while the sum of the same two numbers is placed on bottom. Copy and complete
each Diamond Problem below.
2-41. Solve the following systems of equations using any method. Check your solution if possible.
a. 6x − 2y = 10
3x − y = 2
b. x − 3y = 1
y = 16 − 2x
c. y = 4x + 5
y = −2x – 13
2-42. Write the area of the rectangle below as a sum and as a product.
2-43. Use products and sums or generic rectangles to factor the following expressions.
a. x2 + 9x + 20
b. x2 – 8x + 15
c. y2 – 13y + 30
d. y2 – 3y – 10
2-44. Multiply the expressions below using a generic rectangle. Then verify Casey's pattern (that
the product of one diagonal equals the product of the other diagonal, see problem 2-38).
a. (4x − 1)(3x + 5)
b. (2x − 7)2
2-45. Erica works in a soda-bottling factory. As bottles pass her on a conveyer belt, she puts caps
on them. Unfortunately, Erica sometimes breaks a bottle before she can cap it. She gets paid 4
cents for each bottle she successfully caps, but her boss deducts 2 cents from her pay for each bottle
she breaks.
Erica is having a bad morning. Fifteen bottles have come her way, but she has been breaking some
and has only earned 6 cents so far today. How many bottles has Erica capped and how many has
she broken? Write a system of equations representing this situation and solve the problem.
2-46. Miguel wants to use a generic rectangle to factor 3x2 + 10x + 8. He knows that 3x2 and 8 go
into the rectangle in the locations shown at right. Finish the rectangle by deciding how to place the
10 x-terms. Then write the area as a product.
2-47. Factor each quadratic expression below, if possible. Use a Diamond Problem or generic
rectangle for each one.
a. 2x2 + 5x + 3
b. 2x2 + 7x + 6
c. 2x2 – 9x – 5
d. 3m2 + m − 14
2-48. Solve each equation below for the given variable, if possible.
a. −3(2b − 7) = −3b + 21 − 3b
b. 6 − 2(c − 3) = 12
2-49. Solve the following systems of equations using any method. Check your solution if possible.
a. y = 2x − 3
x + y = 15
b. 3x – y = − 2
6x + 2y = 4
c. y = 2x + 5
y = −2x – 15
2-50. Use generic rectangles or the distributive property to multiply each of the following
expressions.
a. (x + 2)(x − 5)
b. (y + 2x)(y + 3x)
c. (3y − 8)(−x + y)
d. (x − 3y)(x + 3y)
2-51. Factor each quadratic expression below, if possible. Use a Diamond Problem or generic
rectangle for each one.
a. 2x2 + 3x − 5
b. 4x2 + 4x + 1
c. 3x2 + 13x + 4
d. 4x2 − 8x + 3
2-52. Examine the generic rectangle at right. Determine the missing attributes and then write the
area as a product and as a sum.
2-53. Use products and sums or generic rectangles to factor the following expressions.
a. x2 + 12x + 36
b. x2 – 8x + 16
c. y2 – 6y + 9
d. y2 + 20y + 100
2-54. Copy and complete each of the Diamond Problems below. The pattern used in the Diamond
Problems is shown at right.
a.
b.
c.
d.
2-55. Write a system of equations to represent the situation and then solve. Be sure to define your
variable(s) and clearly answer the question.
In Katy’s garden there are 105 ladybugs. They are increasing at two ladybugs per month. There are
currently 175 aphids and the number of aphids is decreasing at three aphids per month. When will
the number of ladybugs and aphids in Katy’s garden be the same?
2-56. SPECIAL QUADRATICS
Use products and sums or generic rectangles to factor the following expressions (if possible). Look
for similarities and differences among the expressions and their corresponding factored forms.
Then sort them into groups based on the patterns you find in their factored forms.
a. x2 – 49
g. x2 − 6x + 9
b. x2 + 2x – 24
h. x2 − 36
c. x2 − 10x + 25
i. x2 − 20x + 100
d. 9x2 + 12x + 4
j. 4x2 + 20x + 25
e. 5x2 − 4x – 1
k. x2 – 4
f. 4x2 – 25
l. 9x2 − 1
2-57. Which of the following quadratic expressions fit the patterns you found in problem 2-56?
Factor each of the following expressions using your new shortcuts, if possible.
a. 25x2 − 1
b. x2 − 5x − 36
c. x2 + 8x + 16
d. 9x2 – 100
2-58. Multiply each pair of polynomials.
a. (2a + b)(a – 3b)
b. (x + 2)(x2 – 2x + 5)
2-59. Solve each equation.
a. 10 – 2(2x + 1) = 4(x – 2)
c. 8a + a − 3 = 6a − 2a − 3
b. 5 – (2x – 3) = –8 + 2x
d. (m + 2)(m + 3) = (m + 2)(m − 2)
8-60. Solve each of the following systems of equations algebraically.
a. y = 4x + 5
y = −2x – 13
b. 2x + y = 9
y = −x + 4
8-61. Find the dimensions of the generic rectangle shown below and write its area as a sum and a
product.
8-62. Write a system of equations to represent the situation and then solve. Be sure to define your
variable(s) and clearly answer the question.
Adult tickets for the school play cost $5 and student tickets cost $3. Thirty more student tickets
were sold than adult tickets. If $1770 was collected, how many of each type of ticket was sold?
2-63. Factor each quadratic expression below, if possible. Use a Diamond Problem or generic
rectangle for each one.
a. 2x2 + 11x + 12
b. 6x2 + x – 2
c. 3x2 + 7x + 2
d. 2x2 − 7x – 15
2-64. Copy and complete each of the Diamond Problems below.
a.
b.
c.
d.