2. 2.1 – Linear and Quadratic Equations
Linear Equations

3. 2.1 – Linear and Quadratic Equations
A quadratic equation is written in the Standard Form,
where a, b, and c are real numbers and 𝑎≠0.
Examples:
(standard form)

4. 2.1 – Linear and Quadratic Equations
Zero Factor Property:
If a and b are real numbers and if ,
then or
Examples:

5. 2.1 – Linear and Quadratic Equations
Zero Factor Property:
If a and b are real numbers and if ,
then or
Examples:

6. 2.1 – Linear and Quadratic Equations
Solving Quadratic Equations:
1) Write the equation in standard form.
2) Factor the equation completely.
3) Set each factor equal to 0.
4) Solve each equation.
5) Check the solutions (in original equation).

7. 2.1 – Linear and Quadratic Equations

8. 2.1 – Linear and Quadratic Equations

9. 2.1 – Linear and Quadratic Equations
𝑃𝑟𝑜𝑑𝑢𝑐𝑡:3∙6=18
𝐹𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 18: 1,18 2,9 3,6
𝐹𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 18 𝑡ℎ𝑎𝑡 𝑐𝑜𝑚𝑏𝑖𝑛𝑒 𝑡𝑜 7: 2,9
3 𝑥 2 −2𝑥+9𝑥−6=0
𝑥 3𝑥−2 +3 3𝑥−2 =0
3𝑥−2 𝑥+3 =0

10. 2.1 – Linear and Quadratic Equations

11. 2.1 – Linear and Quadratic Equations

12. 2.1 – Linear and Quadratic Equations

13. 2.1 – Linear and Quadratic Equations
9 𝑥 2 −3𝑥−12𝑥+4=0
3𝑥 3𝑥−1
−4 3𝑥−1 =0
3𝑥−1
3𝑥−4 =0
3𝑥−1=0
3𝑥−4=0
𝑥= 1 3
𝑥= 4 3

14. 2.1 – Linear and Quadratic Equations
3 𝑥 3 +2 𝑥 2 −12𝑥−8=0
𝑥 2 3𝑥+2
−4 3𝑥+2 =0
3𝑥+2
𝑥 2 −4 =0
3𝑥+2
𝑥−2
𝑥+2 =0
3𝑥+2=0
𝑥−2=0
𝑥+2=0
𝑥=− 2 3
𝑥=2
𝑥=−2

15. 2.1 – Linear and Quadratic Equations

16. 2.1 – Linear and Quadratic Equations
𝐒𝐩𝐞𝐜𝐢𝐚𝐥 𝐄𝐪𝐮𝐚𝐭𝐢𝐨𝐧𝐬
3 4𝑥−11 =−12(3−𝑥)
12𝑥−33 =
−36+12𝑥
−12𝑥 −12𝑥
−33=−36
𝑓𝑎𝑙𝑠𝑒 𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡
𝑛𝑜 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛

17. 2.1 – Linear and Quadratic Equations
𝐒𝐩𝐞𝐜𝐢𝐚𝐥 𝐄𝐪𝐮𝐚𝐭𝐢𝐨𝐧𝐬
3 2𝑥+6−5𝑥 =11𝑥+7−20𝑥+11
6𝑥+18−15𝑥 =
−9𝑥+18
−9𝑥+18=−9𝑥+18
+9𝑥 𝑥
18=18
𝑡𝑟𝑢𝑒 𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡
𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠
𝐼𝑑𝑒𝑛𝑡𝑖𝑡𝑦 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛

18. 2.2 - Formulas
Using the given values, solve for the variable in each formula that was not assigned a value.
Example 1:
Check:

19. 2.2 - Formulas
Example 2: Volume of a Pyramid
LCD: 3

20. 2.2 - Formulas Example 3: Solve for the requested variable.
Area of a Triangle – solve for b
LCD: 2

21. Celsius to Fahrenheit – solve for C
2.2 - Formulas
Example 4: Solve for the requested variable.
Celsius to Fahrenheit – solve for C
LCD: 5

22. 2.2 - Formulas Example 6: Solve for the requested variable.
Solve for v
ℎ=𝑣𝑡−16 𝑡 2
+16 𝑡 𝑡 2
ℎ+16 𝑡 2 =𝑣𝑡
ℎ+16 𝑡 2 𝑡 = 𝑣𝑡 𝑡
ℎ+16 𝑡 2 𝑡 =𝑣

23. 2.2 - Formulas Example 5: Solve for the requested variable.
Solve for x
𝑎𝑥−5=𝑐𝑥−2
−𝑐𝑥 −𝑐𝑥
𝑎𝑥−𝑐𝑥−5=−2
𝑎𝑥−𝑐𝑥=3
𝑥(𝑎−𝑐)=3
𝑥(𝑎−𝑐) 𝑎−𝑐 = 3 𝑎−𝑐
𝑥= 3 𝑎−𝑐

24. 2.3 - Applications Simple Interest Simple Interest = Principal
Interest Rate
Time . .
Interest Rate is stated as a percent and converted to a decimal for calculation purposes.
Time is stated in years or part of a year.
𝑰=𝑷𝑹𝑻

25. 2.3 - Applications
𝑰=𝑷𝑹𝑻
Find the simple interest on a five year loan of $875 at a rate of 7%.
Simple Interest
$875 7% 5 . .
875
0.07 5 . .
61.25 5 .
306.25
$306.25

26. 2.3 - Applications
𝑰=𝑷𝑹𝑻
You invest $8000 in two accounts and earn a total of $323 in interest from both accounts in one year. The interest rates on the accounts were 4.6% and 2.8%. How much was invested in each account?
Principal
Interest Rate
Interest
Account 1 x
𝟒.𝟔%=𝟎.𝟎𝟒𝟔
𝟎.𝟎𝟒𝟔𝒙
Account 2
8000-x
𝟐.𝟖%=𝟎.𝟎𝟐𝟖
𝟎.𝟎𝟐𝟖(𝟖𝟎𝟎𝟎−𝒙)
Total Interest = Account 1 + Account 2
Account 1: $5500
𝟑𝟐𝟑=𝟎.𝟎𝟒𝟔𝒙+𝟎.𝟎𝟐𝟖(𝟖𝟎𝟎𝟎−𝒙)
Account 2: 8000 – 5500
𝟑𝟐𝟑=𝟎.𝟎𝟒𝟔𝒙+𝟐𝟐𝟒−𝟎.𝟎𝟐𝟖𝒙
Account 2: $2500
𝟗𝟗=𝟎.𝟎𝟏𝟖𝒙
𝟓𝟓𝟎𝟎=𝒙

27. 2.3 - Applications
In a right triangle, the length of the longer leg is 7 more inches than the shorter leg. The length of the hypotenuse is 8 more inches than the length of the shorter leg. Find the length of all three sides.
x + 8
𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐 x
(𝒙) 𝟐 + (𝒙+𝟕) 𝟐 = (𝒙+𝟖) 𝟐
x + 7
𝒙 𝟐 + 𝒙 𝟐 +𝟏𝟒𝒙+𝟒𝟗= 𝒙 𝟐 +𝟏𝟔𝒙+𝟔𝟒
− 𝒙 𝟐 −𝟏𝟔𝒙−𝟔𝟒 − 𝒙 𝟐 −𝟏𝟔𝒙−𝟔𝟒
𝒙 𝟐 −𝟐𝒙−𝟏𝟓=𝟎
𝒙=𝟓 𝒊𝒏𝒄𝒉𝒆𝒔
(𝒙+𝟑)
(𝒙−𝟓) =𝟎
𝒙+𝟕=𝟏𝟐 𝒊𝒏𝒄𝒉𝒆𝒔
𝒙+𝟑=𝟎
𝒙−𝟓=𝟎
𝒙+𝟖=𝟏𝟑 𝒊𝒏𝒄𝒉𝒆𝒔
𝒙=−𝟑
𝒙=𝟓

28. 2.3 - Applications
A family paid $26,250 as a down payment for a home. This represents 15% of the selling price. What is the price of the home?
𝑺𝒆𝒍𝒍𝒊𝒏𝒈 𝒑𝒓𝒊𝒄𝒆×𝒅𝒐𝒘𝒏 𝒑𝒂𝒚𝒎𝒆𝒏𝒕 𝒑𝒆𝒓𝒄𝒆𝒏𝒕𝒂𝒈𝒆=𝒅𝒐𝒘𝒏 𝒑𝒂𝒚𝒎𝒆𝒏𝒕
𝒑×𝟎.𝟏𝟓
=𝟐𝟔𝟐𝟓𝟎
𝟎.𝟏𝟓𝒑=𝟐𝟔𝟐𝟓𝟎
𝟎.𝟏𝟓𝒑 𝟎.𝟏𝟓 = 𝟐𝟔𝟐𝟓𝟎 𝟎.𝟏𝟓
𝒑=𝟏𝟕𝟓𝟎𝟎𝟎
𝒑=$𝟏𝟕𝟓,𝟎𝟎𝟎

29. Special Pairs of Angles
2.3 - Applications
Special Pairs of Angles
Complimentary angles: Two angles whose sum is 90°. They are compliments of each other.
Supplementary angles: Two angles whose sum is 180°. They are supplements of each other.
One angle is six more than five times the other angle. What are their measurements if the are supplements of each other? 𝒙
𝒙+𝟓𝒙+𝟔
=𝟏𝟖𝟎
𝟔𝒙+𝟔=𝟏𝟖𝟎
𝟓𝒙+𝟔
𝒙=𝟐𝟗°
𝟔𝒙=𝟏𝟕𝟒
𝟓 𝟐𝟗 +𝟔=𝟏𝟓𝟏°
𝟔𝒙 𝟔 = 𝟏𝟕𝟒 𝟔
𝟐𝟗°+𝟏𝟓𝟏°=𝟏𝟖𝟎°
𝒙=𝟐𝟗

30. 2.3 - Applications
A flower bed is in the shape of a triangle with one side twice the length of the shortest side, and the third side is 30 feet more than the length of the shortest side. Find the dimensions if the perimeter is 102 feet.
x = the length of the shortest side x 2x
2x = the length of the second side
x + 30 = the length of the third side
x + 30
P = a + b + c
102 = x + 2x + x + 30
𝑥 = 18 𝑓𝑒𝑒𝑡
102 = 4x + 30
2(18) = 36 𝑓𝑒𝑒𝑡
102 – 30 = 4x + 30 – 30
= 48 𝑓𝑒𝑒𝑡
72 = 4x →
𝑥=18

31. 2.3 - Applications
The length of a rectangle is 4 less than twice the width. The area of the rectangle is 70. Find the dimensions of the rectangle.
𝑤𝑖𝑑𝑡ℎ=𝑥
𝑙𝑒𝑛𝑔𝑡ℎ=2𝑥−4
𝐴=𝑙×𝑤
2𝑥−4
70= 2𝑥−4 𝑥
70=2 𝑥 2 −4𝑥 𝑥
0=2 𝑥 2 −4𝑥−70
0=2( 𝑥 2 −2𝑥−35)
0=2(𝑥+5)(𝑥−7)
𝑥+5 =0
𝑥−7 =0
𝑥=−5
𝑥=7
𝑤𝑖𝑑𝑡ℎ=𝑥=7
𝑙𝑒𝑛𝑔𝑡ℎ=2𝑥−4=2 7 −4=10

32. 2.3 - Applications
Two cars leave an airport at the same time. One is traveling due north at a rate of 40 miles per hour and the other is travelling due east at a rate of 35 miles per hour. When will the distance between the two cars be 110 miles?
𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆=𝒓𝒂𝒕𝒆 ×𝒕𝒊𝒎𝒆
110 mi.
𝒅 𝒏 =𝟒𝟎𝒕
𝒅 𝒆 =𝟑𝟓𝒕
40 mph
𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐
(𝟒𝟎𝒕) 𝟐 + (𝟑𝟓𝒕) 𝟐 = 𝟏𝟏𝟎 𝟐
35 mph
𝟏𝟔𝟎𝟎𝒕 𝟐 + 𝟏𝟐𝟐𝟓𝒕 𝟐 =𝟏𝟐𝟏𝟎𝟎
𝟐𝟖𝟐𝟓𝒕 𝟐 =𝟏𝟐𝟏𝟎𝟎
𝟐𝟖𝟐𝟓 𝒕 𝟐 𝟐𝟖𝟐𝟓 = 𝟏𝟐𝟏𝟎𝟎 𝟐𝟖𝟐𝟓
𝒕 𝟐 =𝟒.𝟐𝟖𝟑 →
𝒕=𝟐.𝟎𝟕 𝒉𝒓𝒔.

33. 2.3 - Applications Rate Equation: 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒=𝑟𝑎𝑡𝑒 × 𝑡𝑖𝑚𝑒 (𝑑=𝑟𝑡)
It takes Karen 3 hours to row a boat 30 kilometers upstream in a river. If the current was 4 kilometers per hour, how fast would she row in still water?
Rate upstream:
(𝑑=𝑟𝑡)
30= 𝑟 𝑢 (3)
𝑟 𝑢 =10 𝑘𝑝ℎ
Rate in still water:
𝑟 𝑠 =10 𝑘𝑝ℎ+4𝑘𝑝ℎ
𝑟 𝑠 =14 𝑘𝑝ℎ
How long would it take her to row 30 kilometers in still water?
(𝑑=𝑟𝑡)
30=14 𝑡
𝑡=2.14 ℎ𝑟𝑠.
How long would it take her to row 30 kilometers downstream?
(𝑑=𝑟𝑡)
30=(14+4) 𝑡
30=18 𝑡
𝑡=1.67 ℎ𝑟𝑠.

34. 2.4 – Linear Inequalities in One Variable
An inequality is a statement that contains one of the symbols: < , >, ≤ or ≥.
Equations
Inequalities
x = 3
x > 3
12 = 7 – 3y
12 ≤ 7 – 3y
A solution of an inequality is a value of the variable that makes the inequality a true statement.
The solution set of an inequality is the set of all solutions.

35. 2.4 – Linear Inequalities in One Variable

36. 2.4 – Linear Inequalities in One Variable

37. 2.4 – Linear Inequalities in One Variable
Example
Graph each set on a number line and then write it in interval notation. a. b. c.

38. 2.4 – Linear Inequalities in One Variable
Addition Property of Inequality
If a, b, and c are real numbers, then
a < b and a + c < b + c
a > b and a + c > b + c
are equivalent inequalities.
Also,
If a, b, and c are real numbers, then
a < b and a - c < b - c
a > b and a - c > b - c
are equivalent inequalities.

39. 2.4 – Linear Inequalities in One Variable
Example
Solve: Graph the solution set. [

40. 2.4 – Linear Inequalities in One Variable
Multiplication Property of Inequality
If a, b, and c are real numbers, and c is positive, then
a < b and ac < bc
are equivalent inequalities.
If a, b, and c are real numbers, and c is negative, then
a < b and ac > bc
are equivalent inequalities.
The direction of the inequality sign must change when multiplying or dividing by a negative value.

41. 2.4 – Linear Inequalities in One Variable
Example
Solve: Graph the solution set.
The inequality symbol is reversed since we divided by a negative number. (

42. 2.4 – Linear Inequalities in One Variable
Solve: 3x + 9 ≥ 5(x – 1). Graph the solution set.
3x + 9 ≥ 5(x – 1)
3x + 9 ≥ 5x – 5
3x – 3x + 9 ≥ 5x – 3x – 5
9 ≥ 2x – 5
9 + 5 ≥ 2x – 5 + 5
14 ≥ 2x
7 ≥ x
x ≤ 7 [

43. 2.4 – Linear Inequalities in One Variable
Example
Solve: 7(x – 2) + x > –4(5 – x) – 12. Graph the solution set.
7(x – 2) + x > –4(5 – x) – 12
7x – 14 + x > –20 + 4x – 12
8x – 14 > 4x – 32
8x – 4x – 14 > 4x – 4x – 32
4x – 14 > –32
4x – > –
4x > –18
x > –4.5 (

44. Compound Inequalities
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Intersection of Sets
The solution set of a compound inequality formed with and is the intersection of the individual solution sets.

45. Compound Inequalities
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Example
Find the intersection of:
The numbers 4 and 6 are in both sets.
The intersection is {4, 6}.

46. Compound Inequalities
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Example
Solve and graph the solution for x + 4 > 0 and 4x > 0.
First, solve each inequality separately.
x + 4 > 0
4x > 0
and
x > – 4
x > 0 (
(0, )

47. Compound Inequalities
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Example
0  4(5 – x) < 8
0  20 – 4x < 8
0 – 20  20 – 20 – 4x < 8 – 20
– 20  – 4x < – 12
Remember that the sign direction changes when you divide by a number < 0! ( 3 4 5 [
5  x > 3
(3,5]

48. Compound Inequalities
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Example – Alternate Method
0  4(5 – x) < 8
0  4(5 – x)
4(5 – x) < 8
0  20 – 4x
20 – 4x < 8
0 – 20  20 – 20 – 4x
20 – 20 – 4x < 8 – 20
– 20  – 4x
– 4x < – 12
Dividing by negative:
change sign
Dividing by negative:
change sign
x > 3
5  x ( 3 4 5 [
(3,5]

49. Compound Inequalities
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Union of Sets
The solution set of a compound inequality formed with or is the union of the individual solution sets.

50. Compound Inequalities
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Example
Find the union of:
The numbers that are in either set are
{2, 3, 4, 5, 6, 8}.
This set is the union.

51. Compound Inequalities
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Example: Solve and graph the solution for
5(x – 1)  –5 or 5 – x < 11
5(x – 1)  –5 or
5 – x < 11
5x – 5  –5
–x < 6
5x  0
x > – 6
x  0
(–6, )

52. Compound Inequalities
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Example: or