1. Algebra 2 10/23/14 Review for Test 2 (non-calculator section)
What you’ll learn and why… I can learn how to solve the problems in the Practice Test so I can ace Test 2 on Monday. HW: Know the formulas and methods needed to solve each problem in the Practice Test.
Warm-up:
Silently recite the Quadratic Formula 5x.
Silently recite the formula to find the x-coordinate of the vertex of a quadratic equation in standard form 5x.

2. 1. What is the product of 𝟒+𝟐𝒊 and 𝟒−𝟐𝒊 written in standard form?20
16 + 4i
D. 16 – 4i
Notes:
4+2𝑖 and 4−2𝑖 are complex conjugates.
The product of complex conjugates is always a real number. From this information, choices C and D should be eliminated.
Rule or Formula to multiply complex conjugates:
𝒂+𝒃𝒊 𝒂−𝒃𝒊 = 𝒂 𝟐 + 𝒃 𝟐
𝟒+𝟐𝒊 𝟒−𝟐𝒊 = 𝟒 𝟐 + 𝟐 𝟐 =𝟏𝟔+𝟒=𝟐𝟎
The answer is B 20.
Your Turn: Answer Practice Test #1.

3. Your Turn: Answer Practice Test #2
2. Simplify 𝟑+𝟒𝒊 −(−𝟐−𝟓𝒊).
A. −𝟏+𝟗𝒊
B. 𝟏−𝒊
C. 𝟓+𝟗𝒊
D. 𝟓−𝒊
Notes:
To subtract a complex number, add its opposite.
One Method:
3+4𝑖 − −2−5𝑖 = 3+4𝑖 +(2+5𝑖)
=3+4𝑖+2+5𝑖
=𝟓+𝟗𝒊
Answer: C
Your Turn: Answer Practice Test #2

4. 3. Find the domain and range of each function.B D: R: C D: R: D D: R:
All real numbers
All real numbers
All real numbers
All real numbers
𝒚≥𝟎
𝒚 ≤ 𝟎
𝒚 ≤ 1
𝒚≥−𝟐
Notes:
The domain of a function is the set of all possible x values.
The range of a function is the set of all possible y values.

5. Your Turn: Answer Practice Test #4.
4. Which of the following is not true of the function 𝒇 𝒙 = 𝒙 𝟐 −𝟐𝒙+𝟓?
2. Since 𝑎>0, the parabola opens up, hence the function has a minimum.
The minimum is the y-coordinate of the vertex.
To find the minimum, first find the x-coordinate of the vertex. Use the formula for the axis of symmetry to find the x-coordinate, 𝒙= −𝒃 𝟐𝒂
The y-intercept is 5.
The minimum value is 4.
The axis of symmetry is 𝒙=𝟏.
The vertex is at (1, 5).
Notes:
1. The function 𝑓 is in standard form, 𝑓 𝑥 =𝑎 𝑥 2 +𝑏𝑥+𝑐.
2. The y-intercept is at x=0, so y−intercept=𝑐=5.
Hence, Choice A is true.
Since a = 1 and b = -2, 𝒙= −(−𝟐) 𝟐(𝟏) =𝟏
3. To find the y-coordinate of the vertex, substitute the value of x into the original function.
𝒚= (𝟏) 𝟐 −𝟐 𝟏 +𝟓=𝟏−𝟐+𝟓=−𝟏+𝟓=𝟒
Hence, Choices B and C are true.
The statement which is not true is D. The vertex is at (1, 4)
Your Turn: Answer Practice Test #4.

6. Your Turn: Answer Practice Test #5
5. If 𝒙 𝟐 −𝟔𝒙+𝟐= 𝒙−𝒉 𝟐 +𝒌, what is the value of 𝒌? -7
-11 7 11
Notes:
1. 𝑥−ℎ 2 +𝑘 is the vertex form of a quadratic expression.
2. In the vertex form, (h, k) is the vertex and 𝑘 is the y-coordinate .
Possible method:
Find the coordinates of the vertex by using the formula for the axis of symmetry or the x-coordinate:
𝒙= −𝒃 𝟐𝒂 = −(−𝟔) 𝟐(𝟏) = 𝟔 𝟐 =𝟑
Next, solve for the y-coordinate of the vertex by substituting 𝑥=3 into the original expression.
𝒚= (𝟑) 𝟐 −𝟔 𝟑 +𝟐=𝟗−𝟏𝟖+𝟐=−𝟗+𝟐=−𝟕
Since 𝒚=−𝟕, 𝒕𝒉𝒆𝒓𝒆𝒇𝒐𝒓𝒆 𝒌=−𝟕. The correct answer is A.
Alternative Method: You could use completing the square to convert from standard form to vertex form.
Your Turn: Answer Practice Test #5

7. 6. Find the roots of 𝟏𝟔 𝒙 𝟐 +𝟑=𝟎. A. 𝑥=± 3 4 Solution:
Goal: Rewrite the equation so that it is in the form
𝒙 𝟐 =𝒄.
𝒙=± 𝒄
B. 𝑥=± −3 16
C. 𝑥=± 𝑖 3 4
16 𝑥 2 +3=0
D. 𝑥=±
16 𝑥 2 =−3
Subtract 3 from each side
𝑥 2 = −3 16
Divide each side by 16.
Note:
Eliminate answers that are not in simplest form. Choices A and B are eliminated because they have a radical in the denominator. The choices left are C and D. C has nonreal solutions while D has real solutions.
𝑥=± −3 16
𝒙 𝟐 =𝒄.
𝒙=± 𝒄
𝑥=± −
𝐐𝐮𝐨𝐭𝐢𝐞𝐧𝐭 𝐑𝐮𝐥𝐞 𝐨𝐟 𝐑𝐚𝐝𝐢𝐜𝐚𝐥𝐬
𝒎 𝒏 = 𝒎 𝒏
𝒙=± 𝒊 𝟑 𝟒
Simplify.

8. 7. Factor 𝟐𝟓 𝒙 𝟐 +𝟔𝟒.
Notes:
25 𝑥 is a Sum of Two Squares. The factors of a Sum of Two Squares are two complex conjugates. Eliminate Choices A and B.
A. 5𝑥+8 (5𝑥−8)
B. (5𝑥+8)(5𝑥+8)
C. (5𝑥−8𝑖)(5𝑥+8𝑖)
D. (5𝑥+8𝑖)(5𝑥+8𝑖)
The rule is:
𝒂 𝟐 + 𝒃 𝟐 =(𝒂+𝒃𝒊)(𝒂−𝒃𝒊)
𝟐𝟓𝒙 𝟐 +𝟔𝟒=(𝟓𝒙+𝟖𝒊)(𝟓𝒙−𝟖𝒊)
The answer is C.
Alternative Method: Work Backwards
Start from the answer choices. Use FOIL to find which factors will multiply to get the product 𝟐𝟓𝒙 𝟐 +𝟔𝟒.
𝟓𝒙−𝟖𝒊 𝟓𝒙+𝟖𝒊 =𝟐𝟓 𝒙 𝟐 +𝟒𝟎𝒙𝒊−𝟒𝟎𝒙𝒊−𝟔𝟒 𝒊 𝟐 =𝟐𝟓 𝒙 𝟐 −𝟔𝟒 −𝟏 =𝟐𝟓 𝒙 𝟐 +𝟔𝟒
Your Turn: Answer Practice Test #7.

9. 8. Solve 𝒙 𝟐 +𝟖𝟏=𝟎 over the set of complex numbers.
A. ±9
B. ±9𝑖
C. ±81
D. ±81𝑖
Notes:
The goal is to rewrite the equation in the form:
𝒙 𝟐 =𝒄
𝒙=± 𝒄
𝑥 2 +81=0
𝑥 2 =−81
Subtract 81 from each side.
𝑥=± −81
𝒙 𝟐 =𝒄
𝒙=± 𝒄
𝑥=±9𝑖
Simplify. 𝒊= −𝟏
The answer is B.
Your Turn: Answer Practice Test #8.

10. Your Turn: Answer Practice Test #9.
Function A and Function B are continuous quadratic functions.
Function A Function B
𝒇 𝒙 = 𝒙 𝟐 +𝟒𝒙−𝟓
Which function has a greater negative x-intercept?
Function A
Function B
The x-intercepts are equal.
Solution: You need to find the negative x-intercepts of both function and then compare to find the greater number.
The negative x-intercept of Function B is -1.
Use factoring to find the x-intercepts of Function A. (Other methods may be used.)
𝒙 𝟐 +𝟒𝒙−𝟓=𝟎
𝒂=𝟏, 𝒃=𝟒, 𝒄=−𝟓
𝒙+𝟓 𝒙−𝟏 =𝟎
Look for factors of c whose sum is b. Use 5 and -1
𝒙+𝟓=𝟎 𝒐𝒓 𝒙−𝟏=𝟎
Set each factor equal to 0.
𝒙=−𝟓 𝒐𝒓 𝒙=𝟏
Solve for x.
The negative x-intercept of Function A is -5. Since −𝟏>−𝟓, the answer is B.
Your Turn: Answer Practice Test #9.

11. 10. Solve the equation 𝟓 𝒙 𝟐 +𝟓𝒙=𝟔𝟎 by factoring.
𝒙=−𝟒 𝒐𝒓 𝒙=𝟑
𝒙=𝟒 𝒐𝒓 𝒙=−𝟑
Steps:
𝟓 𝒙 𝟐 +𝟓𝒙=𝟔𝟎
𝒙 𝟐 +𝒙=𝟏𝟐
Divide each side by the GCF, 5.
𝒙 𝟐 +𝒙−𝟏𝟐=𝟎
Subtract 12 from each side.
(𝒙+𝟒)(𝒙−𝟑)=𝟎
Look for factors of -12 whose sum is 1. Use 4 and -3.
𝒙+𝟒=𝟎 𝒐𝒓 𝒙−𝟑=𝟎
Set each factor equal to 0.
𝒙=−𝟒 𝒐𝒓 𝒙=𝟑
Solve for x.
The answer is C.
Alternative Method: Work Backwards
Substitute the answer choices into the given equation. See which solutions satisfy the given equation.
Your Turn: Answer Practice Test #10.

12. Your Turn: Answer Practice Test #11.
11. Which of the following functions has its vertex below the x-axis?
A. 𝒇 𝒙 =− 𝒙 𝟐
B. 𝒇 𝒙 =−𝟓 (𝒙+𝟒) 𝟐 +𝟑
C. 𝒇 𝒙 = 𝒙 𝟐 +𝟐
D. 𝒇 𝒙 =𝟐 (𝒙−𝟏) 𝟐 −𝟐
Solution: To determine which vertex is below the x-axis, graph the vertex of each function.
Vertex Form: 𝑓 𝑥 =𝑎 (𝑥−ℎ) 2 +𝑘
Vertex is at (h, k)
Vertex of A: (0, 0)
Vertex of C: (0, 2)
Vertex of B: (-4, 3)
Vertex of D: (1, -2)
The answer is D.
Is there a pattern? What pattern do you see?
Given the vertex (h, k), when k < 0, then the vertex is below the x-axis.
Your Turn: Answer Practice Test #11.

13. 12. What is the equation of the parabola shown?
Notes:
1. The parabola opens up, therefore a > 0.
B. 𝑓 𝑥 =− 1 4 𝑥 2
C. 𝑓 𝑥 = 1 4 𝑥 2
Eliminate Choices A and B.
D. 𝑓 𝑥 = 4𝑥 2
2. USE NICE POINTS!
The point (1, 4) is on the parabola. Check by substituting x = 1 and y = 4 into the remaining equations C and D.
C. 𝑓 1 = 1 4 (1) 2 = 1 4
D. 𝑓 1 =4 (1) 2 = 4
The answer is D.
Your Turn: Answer Practice Test #12.

14. 13. What is the solution set of 𝟑 𝒙 𝟐 +𝟖𝒙>−𝟑𝒙−𝟔?
B. −3≤𝑥≤− 2 3
C. 𝑥<−3 𝑜𝑟 𝑥>− 2 3
D. 𝑥>−3 𝑜𝑟 𝑥>− 2 3

15. 13. What is the solution set of 𝟑 𝒙 𝟐 +𝟖𝒙>−𝟑𝒙−𝟔?
B. −3≤𝑥≤− 2 3
C. 𝑥<−3 𝑜𝑟 𝑥>− 2 3
D. 𝑥>−3 𝑜𝑟 𝑥>− 2 3
STEPS
1. Write the original inequality.
𝟑 𝒙 𝟐 +𝟖𝒙>−𝟑𝒙−𝟔
2. Set one side of the inequality to 0.
(Add 3x and 6 to each side.)
𝟑 𝒙 𝟐 +𝟖𝒙+𝟑𝒙+𝟔>𝟎
3. Combine like terms.
𝟑 𝒙 𝟐 +𝟏𝟏𝒙+𝟔>𝟎
4. Change the inequality sign to equal sign.
𝟑 𝒙 𝟐 +𝟏𝟏𝒙+𝟔=𝟎
5. Solve the quadratic equation to find the zeros or x-intercepts.
𝒙=−𝟑 𝒐𝒓 𝒙=− 𝟐 𝟑
6. Mark the x-intercepts on a numberline. The numberline is now divided into three intervals: A, B, and C. Sketch the parabola.
7. Determine in which interval/s the solutions lie. Write the solution set using the correct inequality symbols.
8. Check by using test values.
𝐴𝑡 𝑥=−4: 3 − −4 +6=48−44+6= >0
𝐴𝑡 𝑥=0: 3 (0) = >0 A B C
− 𝟐 𝟑 −𝟑
The solutions lie in intervals A and C.
𝒙<−𝟑 𝒐𝒓 𝒙 >− 𝟐 𝟑

16. 14. Which of the following quadratic equations has no real roots?
B. 𝟓 𝒙 𝟐 −𝟑𝒙−𝟒=𝟎
Write in standard form.
𝑎=5, 𝑏=−3, 𝑐=−4
5 𝑥 2 +3𝑥−4=0
𝑏 2 −4𝑎𝑐= −3 2 −4 5 −4
>𝟎
𝑎=5, 𝑏=3, 𝑐=−4 + +
𝑏 2 −4𝑎𝑐= −4 5 −4
>𝟎 + +
D. 𝟓 𝒙 𝟐 +𝟑𝒙=−𝟒
C. 𝟓 𝒙 𝟐 =𝟑𝒙
Write in standard form.
Write in standard form.
5 𝑥 2 +3𝑥+4=0
5 𝑥 2 −3𝑥=0
𝑎=5, 𝑏=−3, 𝑐=0
𝑎=5, 𝑏=3, 𝑐=4
𝑏 2 −4𝑎𝑐= −4 5 4
<𝟎
𝑏 2 −4𝑎𝑐= −3 2 −4 5 0
>𝟎 - + + +
Note:
Use the discriminant, 𝒃 𝟐 −𝟒𝒂𝒄, to find out which quadratic equation has no real roots.
If 𝒃 𝟐 −𝟒𝒂𝒄 > 0, the equation has 2 real solutions.
If 𝒃 𝟐 −𝟒𝒂𝒄 = 0, the equation has 1 real solution.
If 𝒃 𝟐 −𝟒𝒂𝒄 < 0, the equation has 2 nonreal solutions. 9 80

17. Your Turn: Answer Practice Test #15.
15. Find the vertex of 𝒚= 𝟐𝒙 𝟐 −𝟒𝒙+𝟓 and state if it is a maximum or minimum.
Notes:
Since a > 0, the parabola opens up and it has a minimum value.
(1, 3); maximum
(1, 3); minimum
(3, 1); maximum
(3, 1); minimum
Eliminate Choices A and C.
To find the x-coordinate of the vertex, use the formula 𝒙= −𝒃 𝟐𝒂 .
𝒙= −𝒃 𝟐𝒂 = −(−𝟒) 𝟐(𝟐) = 𝟒 𝟒 =𝟏
The answer is B.
Your Turn: Answer Practice Test #15.