Algebra Q3 Exam Review
Name: __________________________
1) Factor and use the Zero Product Property to find the roots of the following quadratic equations.
a. 0 = x2 − 7x + 12
b. 0 = 6x2 − 23x + 20
c. 0 = x2 − 9
d. 0 = x2 + 12x + 36
2) Graph y = x2 − 2x. Identify the roots, y-intercept, x-intercepts, and the vertex.
3) Find the coordinates of the y-intercept and x-intercepts of y = x2 − 2x − 15. Show all of the work that you
used to find these points.
4) Determine the point of intersection for the equations below without graphing. Show all work.
y= 5x +7
3x + y =9
5) In order to solve x2 + 8x – 3 = y by completing the square, what value must be added to both sides?
6) Multiple Choice: Jenny is solving the equation x2 – 4x = -10. Which number should she add to both
sides of the equation to complete the square (after the equation has been rewritten in standard form)?
A. -6
B. -5
C. 4
D. 9
7) Factor each of the following completely:
a. 9x2 - 4
b. x2 -10x + 25
c. 3x2 + 6x
d. 2x2 + 6x – 56
8) Put each equation into vertex (graphing) form. DO NOT SOLVE!
a. x2 + 10x + 8 = y
b. 2x2 – 8x + 7 = y
9) Use the Zero Product Property to calculate the roots of the polynomials below.
a.
4x(2x-5)
b.
3x2 – x –10
c.
x2- 6x + 9
d.
2(x+4)(x-3)
10) Solve the equations for x. Leave your answer as a simplified radical if needed.
a.
4x2 – 12x = 0
b.
9x2 – 36 = 0
c.
x2 – 8x +15 = 0
d.
y= 2(x+3)2 – 10
e.
y = x2 + 6x +5
f.
-2(x-2)2 = -64
h.
2x2 = 36
i. 0.5x2 = 0.2x +0.1
g.
2x2 –7x = 15
11) What is the equation for the following parabola?
12)
Multiple Answer: Which equation(s) has a
parabola that has x-intercepts of (3,0) and
(7,0), a vertex of (5, -8) and a y-intercept of
(0, 42)?
a) y= 2x2-20x +42
b) y= 2(x -5)2-8
c) y= 3(x-3)(x-7)
d) y= 5(x-8)2 +42
13) A bird sitting at the edge of a cliff sees a fish several feet below the surface of the water in a lake touching
the cliff. The bird dives to catch the fish. The path of the dive follows a parabolic curve given the function
y = x2-12x +32. Imagine that the x-axis runs along the surface of the lake and the y-axis runs along the face of
the cliff. How far below the surface of the water does the bird have to dive to catch the fish?
14) Write an inequality that represents the graph at right.
15) Is the point (0, 4) a solution to the system of inequalities
below? Justify your answer.
16) Brian was holding a ballroom dance. He wanted to make sure girls would come, so he charged boys $5 to
get in but girls only $3. The 45 people who came paid a total of $175. How many girls came to the dance?
17) Factor these quadratic expressions completely, if possible.
a. x2 + x – 30
b. −3x2 + 23x2 − 14x
c. 2x2 − 5x + 4
d. 6x3 + 10x2 − 24x
18) Solve each inequality below for the given variable.
a. 4x − 3 ≥ 9
b. 3(t + 4)< 5
c.
d. 5x + 4 > − 3(x − 8)
20) Solve each quadratic equation using the specified method.
a. The Quadratic Formula
0 = 3x2 + 4x – 7
b. Factoring
x2 − 3x − 18 = 0
c.Completing the square
d. Using a graph
2
x + 4x + 1 = 0
2x2 + 5x − 12 = 0
21) Given the quadratic function f(x) = (x − 1)2 − 4:
a. State the location of the vertex.
b. Determine the x-intercepts.
c. Sketch a graph of the function.
22) Graph the system of inequalities below on graph paper.
23) Lew says to his granddaughter Audrey, “Even if you tripled your age and added 9, you still wouldn’t be as
old as I am.” Lew is 60 years old. Write and solve an inequality to determine the possible ages of Audrey.
24) Factor each polynomial.
a. x2 − x – 56
b. 3x2 − 4x + 1
c. 2x3 + x2 + x
d. 2x2 − 50
25) Solve each quadratic equation using any method. a. 2x2 − x − 5 = 0
b. 4x2 = 4x – 1
26) For the quadratic function f(x) = (x − 3)2 + 4:
a. Identify the vertex and tell if it the maximum or minimum point of the function.
b. Why does (x − 3)2 + 4 = 0 have no real solutions?
31) Make a table and graph the inequality y  x  2 .
32). Graph the following inequalities on separate axes.
a.
y  x 2  3x  1
b.
y  x 1
33) Draw a graph of the solution region for the system of inequalities below.
yx4
y   54 x  6
34) Determine the number of solutions for each quadratic equation below by first completing the square. Explain how
you can determine the number of solutions once the equation is written in perfect square form.
a.
x2 + 6x + 8 = 0
b.
x2 – 6x + 9 = 0
35) Solve by using any method.
a.
t 2  8t  10  0
b.
p2  6 p  1  0
c.
x2 + 6x + 8 = 0
d.
3x2 + 6x = 0
36) Decide whether each of the following points is on the graph of y = –2x2 + 5x – 1. Explain how you know.
a.
(–2, –19)
b.
(2, 19)
b.
x3 + 4x2 + 4x
37) Solve 100  x 2  2  x .
38) Factor each expression below.
a.
9x2 + 24xy + 16y2
39) This parabola shows an equation which could be considered a “special quadratic”. What equation is it and
why is it considered a special quadratic?
40) This parabola shows an equation which could be considered a “special quadratic”. What equation is it and why is it
considered a special quadratic?
41) One type of “special quadratic” is called a “perfect square trinomial”. Why is this considered a “special quadratic”?
Refer to both the equation itself and the graph of the parabola in your answer.
42) One type of “special quadratic” is called a “difference of squares”. Why is this considered a “special quadratic”?
Refer to both the equation itself and the graph of the parabola in your answer.
43) Build a rectangle for each of the following expressions using Algebra Tiles. Then sketch the rectangle on your
paper. Find its dimensions and write its area as a product and a sum.
a.
3x2 + 3x
b.
x2 + xy + y + 3x + 2
c.
2x + 2 + 2xy + 2y + 2x + 2
d.
x2 + 4x + 5
44) Jamie threw a softball that traveled along a path described by the parabola y = -x2 + 10x, where y = the
height of the softball in feet and x = the horizontal distance in feet that the ball has traveled from Jamie. On
separate graph paper, graph the path of the softball.
a.
Where does the softball hit the ground? How can you tell?
b.
Find the vertex. What information does the vertex tell you?
c.
What is the ball’s horizontal distance from Jamie when it is 24 feet up in the air? Does this
make sense? Explain.
d.
Does any of your data not make sense? Explain.
45) Zoe likes to make pancakes. She can flip a pancake in the air and have it land in the frying pan. The
motion of the pancake is represented by the equation y = –9x2 + 9.8x, where x represents the number of
seconds after Zoe flips the pancake in the air, and y is the height of the pancake above the frying pan.
a.
How long is the pancake in the air? How can you tell?
b.
At what time does the pancake stop going up and start coming down?
c.
How high is the pancake at the top of its flight?
46) BUDGETING: Satchi found a used bookstore that sells pre-owned videos and CDs. Videos cost $9 each and CD’s cost
$7 each. Satchi can spend a maximum of $35. Write an inequality to represent her situation.
Let x represent _________________
Let y represent _________________
47) Students in a ninth grade class measured their heights, h, in centimeters. The height of the shortest student was
more than 155 cm, and the height of the tallest kid was 190cm. Write an inequality to represent the heights.
48) Tamara has a cell phone plan that charges $0.07 per minute plus a monthly fee of $19.00. She budgets $29.50 per
month for her cell phone expenses. What is the maximum number of minutes Tamara could use her phone and stay
within her budget? Write an inequality and solve.
49) An online music club has a one-time registration fee of $13.95 and charges $0.49 to buy each song. If Emma had $50
to join the club and buy songs, what is the maximum number of songs she can buy? Write an inequality and solve.
51.
50.
Answers:
1. [a: x = 4 and 3, b: x=5/2 and 4/3, c: x= 3 and -3, d: x=-6]
4. [ (1/4, 8 ¼) ]
5. [ 19 ]
6. [A]
8. [ a: y= (x+5)2 -17, b: y= 2(x-2)2 -1 ]
3. [x-intercepts: (5,0) and (-3,0), y-intercept: (0, -15)]
7. [a: (3x-2)(3x+2), b: (x-5)(x-5) or (x-5)2 , c: 3x(x+2), d: 2(x-4)(x+7)]
9. [ a: x=0 and 5/2, b: x= 5/3 and 2, c: x = 3, d: x= -4 and 3]
10. [a: x=0 and 3, b: x=2 and -2, c: x=3 and 5, d: x = ±√𝟓 -3, e: no solution, f: x = = ±𝟒√𝟐 + 2, g: x= 5 and -3/2, h: x =
±𝟑√𝟐, i: x = –0.29 and 0.69 ]
11. [y= -1/2(x-3)(x+1)]
14. [y > 1/2x -2]
12. [A and B]
15 [yes]
13. [4 feet below the surface of the water]
16. [25 girls]
17. a: (x-5)(x+6); b: -x(x-7)(3x-2); c: Not Factorable; d:2x(x+3)(3x-4)
20. a: x = 1 and -7/3; b: x= 6 and -3; c: x = ±√𝟑 – 2; d: x= -4 and 7/4
23. 0 < 3x + 9 < 60, 0< x < 27
25. a: x=
𝟏 ±√𝟒𝟏
;
𝟒
b: x= ½
18. a: x≥3; b: t< -7/3; c: y<28; d: x>2.5
21. a: (1, -4); b: x = 3 and -1
24. a: (x-8)(x+7); b: (3x-1)(x-1); c: x(2x-1)(x-1); d. 2(x-5)(x+5)
26. a: (3,4); b: there is a negative in the radical.
31.
y
33.
2
2
x
–
2
34. a: (x + 3)2 = 1, two solutions; b: (x – 3)2 = 0, one solution ]
35. [ a: 4  6 , b: 3  8 c: x = –4 and –2, d: x = 0 and –2 ]
36. [ a: yes, b: no ]
37. [ x = 6 ]
38. [ a: (3x + 4y)(3x + 4y), b: (x)(x + 2)(x + 2)
39. [ y=(x–2)(x–2) perfect square trinomial; there is only one root ]
40. [ y=(x–3)(x+3) or y= x2-9 : difference of squares; there is no middle term ]
41. [ A perfect square trinomial is the square of a binomial and it only has one root or x-intercept ]
42. [ A difference of squares quadratic is always in the form of y=(a+b)(a–b) and in sum form y= a2–b2. The graph of
this type of parabola will have roots in which the absolute values of the roots are equal. (5 and –5, for example) ]
43. [ a: 3x2 + 3x = 3x(x + 1), b: x2 + xy + y + 3x + 2 = (x + 1)(y + x + 2), c: 2xy + 2y + 4x + 4 = (2x + 2)(y + 2), d: x2 + 4x + 5
cannot be made into a rectangle because there is one too many units (needs 4 instead of 5) ]
44. [ a: 10 feet away from her, b: (5, 25); the ball is at its highest point of 25 feet when it is a horizontal
distance of 5 feet from Jamie; c: at 4 feet and at 6 feet [the points (4, 24) and (6, 24)], yes, it makes sense,
because the ball is 24 feet off the ground two times, on the way up and on the way down; d: the points
beyond x = 10 don’t make sense because the represent negative (below ground) height ]
45. [ a: 0 seconds and about 1.1 seconds, about 1.1 seconds; b: after about 0.5 seconds; c: about 2.7feet ]
46. [ 9x + 7y ≤ 35; if x represents number of videos and y represents number of cd’s. ]
47. [ 155< h ≤ 190]
48. [0.07x +19 ≤ 29.50; 0≤ x≤150; maximum minutes would be 150]
49. [0.49x + 13.95 ≤ 50, 0≤x≤72.57; maximum songs would be 72]
50. [4]
51. [4]