IM 9 Advanced Problem Sets Problem Set 15: Diamond Problems
1) Consider the expression (𝑥 + 5)(𝑥 + 4).
a) Expand and simplify (𝑥 + 5)(𝑥 + 4).
b) Where does the 9 come from in your answer? Where does the 20 come from?
c) Create a diamond problem that illustrates your explanation to part b.
2) With your solution to 3c in mind, make and complete Diamond Problems for the following:
a) (𝑥 + 5)(𝑥 − 2)
b) (𝑥 − 7)(𝑥 + 3)
c) (𝑥 + 9)(𝑥 + 7)
3) Again thinking about problem 3, make and complete Diamond Problems for the following:
a) 𝑥 ! + 3𝑥 − 10
b) 𝑥 ! − 4𝑥 − 21
c) 𝑥 ! + 16𝑥 + 63
4) Notice that the diamond problems you set up for 4a and 5a are the same! What does this tell you about
the expressions in 4a and 5a? Answer the same question for parts b) and c) by creating a valid
equation.
Problem Set 16: Simplifying Exponents, Geometry Review
1) Use the diagram on the right to find the value of x and y.
3𝑥 ! 𝑦 !
5(3𝑥𝑦 ! )!
34𝑥𝑦 !! 𝑥 !!
2𝑦 !!
2) ∠𝐴𝐵𝐶 and ∠𝐶𝐵𝐷 are complimentary angles, where 𝑚∠𝐴𝐵𝐶 = 𝑥 + 4 and 𝑚∠𝐶𝐵𝐷 = 𝑦 − 3. Find x
and y if 𝑚∠𝐴𝐵𝐶 is twice the size of 𝑚∠𝐶𝐵𝐷. Draw a simple diagram for help if needed.
3) Find the value of x and y in the diagram below, where F is
the midpoint of DE.
IM 9 Advanced Problem Sets Problem Set 17: Diamond Problems, Statistics Review
1) Make and complete a Diamond Problem for each of the following:
a) 𝑥 ! + 8𝑥 − 20
b) 𝑥 ! − 4𝑥 − 32
c) 𝑥 ! + 9𝑥 + 14
2) Use your answers from problem 1 to write each in the form 𝑥 + 𝑎 𝑥 + 𝑏 .
3) Following the steps from problems 1 and 2, write each of the following “quadratic polynomials” in the
form 𝑥 + 𝑎 𝑥 + 𝑏 .
a) 𝑥 ! + 3𝑥 − 54
b) 𝑥 ! − 4𝑥 − 21
c) 𝑥 ! + 7𝑥 + 12
4) Find the mean, median, and mode of the following data set.
3, 6, 2, 7, 5, 3, 7, 5, 3
Problem Set 18: Factoring, Introduction to Special Right Triangles
1) Factor the following “quadratic polynomials”, leaving your answer in the form (x + a)(x + b). Use a
Diamond Problem for help if necessary.
a) x2 + 5x – 24
b) x2 - 7x + 12
c) x2 + 15x + 54
2) In the diagrams below a square has one of its diagonals drawn in to create two triangles.
a) What are the measures of all the angles in a given diagram?
b) Find the value of x and y for each diagram. LEAVE YOUR ANSWER IN SIMPLE RADICAL
FORM.
i)
ii)
iii)
a 7 y y y IM 9 Advanced Problem Sets Problem Set 19: More Factoring, Continuation of Special Right Triangles
1) In the diagrams below an equilateral triangle has been bisected to form two triangles.
a) What are the measures of all the angles in a given diagram?
b) Find the value of x and y for each diagram. LEAVE YOUR ANSWER IN SIMPLE RADICAL
FORM.
i)
ii)
10 iii)
2a 2) For each of the following equations give BOTH valid values of x. Use Diamond Problems if necessary.
a) x2 + 9x - 10 = 0
b) x2 - 16x + 48 = 0
c) x2 - 8x – 33 = 0
Problem Set 20: Continuation of Special Right Triangles, Solving Systems
in a Geometric Context
1*) Solve for x.
a) 𝑥 6 = 18
b) 𝑥 8 = 72
2) a) In the diagram on the right a square has one of its diagonals drawn in to create
two triangles. Find the values of x and y in terms of b.
b y b) In the diagram on the right an equilateral triangle has been bisected to form
two triangles. Find the values of x and y in terms of b.
2b 3) It is known that ∠AOC and ∠COD form a linear pair, and that 𝑂𝐵 bisects ∠AOC. Furthermore, it is
known that ∠AOB = (6x + 5y)°, ∠BOC = (2x - 10y)° and ∠COD = (4x + 5y)°.
a) Draw an accurate and appropriately labeled diagram to illustrate the given information.
b) Find the values of x and y.
c) Find the measure of angles ∠AOB, ∠BOC and ∠COD.
IM 9 Advanced Problem Sets Problem Set 21: Still More Factoring, Continuation of Special Right
Triangles, Solving Sytems in a Geometric Context
1) For each of the following equations give BOTH valid values of x.
a) x2 + 9x + 14 = 0
b) x2 - 17x + 72 = 0
c) x2 - 7x – 18 = 0
2) Solve for x and y in the familiar diagrams below. LEAVE YOUR ANSWER IN SIMPLE RADICAL
FORM.
a)
b) c) 12 12 y 𝟑√𝟐 x b y IM 9 Advanced Problem Sets Answers!
Problem Set 15
1) a) x2 + 9x + 20
2)
b) 4 + 5 = 9, (4)(5) = 20
c)
3)
4) The answers are the same!
x2 + 9x + 20 = (x + 5)(x - 2),
x2 - 4x - 2 = (x + 3)(x - 7), x2 + 16x + 63 = (x + 9)(x + 7)
Problem Set 16
1) x = 15, y = 4
2) x = 56, y = 33
3) x = 4, y = 2
Problem Set 17
1)
2) a) (x + 10)(x - 2) b) (x + 4)(x - 8)
3) a) (x + 9)(x - 6)
b) (x + 3)(x - 7)
4) mean = 4.56, median = 5, mode = 3
Problem Set 18
1) a) (x – 3)(x + 8)
2) a) 45°, 45°, 90°
c) (x + 2)(x + 7)
c) (x + 3)(x + 4)
b) (x – 3)(x - 4)
c) (x + 6)(x + 9)
b) i) 𝑥 = 4 2, 𝑦 = 4 ii) 𝑥 = 7 2, 𝑦 = 7 iii) 𝑥 = 𝑎 2, 𝑦 = 𝑎
Problem Set 19
1) a) 30°, 60°, 90° b) i) 𝑥 = 2, 𝑦 = 2 3 ii) 𝑥 = 5, 𝑦 = 5 3 iii) 𝑥 = 𝑎, 𝑦 = 𝑎 3
2) a) x = -10 or x = 1
b) x = 12 or x = 4
c) x = 11 or x = -3
Problem Set 20
1*) a) 𝑥 = 3 6
2) a) 𝑥 = 𝑏, 𝑦 = 𝑏 2
3) a) get teacher approval
b) 𝑥 = 18 2
!
!
b) 𝑥 = ! , 𝑦 = ! 3
b) x = 15, y = -4
Problem Set 21
1) a) x = -7 or x = -2
2) a) 𝑥 = 12 2, 𝑦 = 12
b) x = 8 or x = 9
b) 𝑥 = 6, 𝑦 = 6 3
c) ∠AOB = 70°, ∠BOC = 70°, ∠COD = 40°
c) x = 9 or x = -2
c) 𝑦 = 3 2, 𝑥 = 6