Name ____________________________________________________________

Basic ALGEBRA 2

SUMMER PACKET

This packet contains Algebra I topics that you have learned before and should be familiar with coming into Algebra II. We will use these concepts on a regular basis throughout Algebra II. There are a few example problems at the beginning of each section, followed by practice problems for you to complete. You are expected to know how to do these problems and to have this packet done before returning to school in the fall. You will have a quiz on the summer packet when you get back to school. You will have approximately one week to check your answers, which are included in the back of the packet, and ask questions before taking several mini-quizzes on the material. These quizzes will be used to assess your understanding of all the concepts in this summer packet.

Have a great summer!!

2.

Table of Contents Glossary of Terminology and Formulas Page 3 - 4

Factoring Polynomials Page 5 – 7

Solving Quadratic Equations Page 8 - 9

Solving Equations for a Variable Page 10

Solving Systems of Equations in Two Variables

Page 10 - 12

Exponents Page 13

Radicals Page 14

Multiplying Polynomials Page 15

Writing the Equation of a Line Given Two Points

Page 16

Graphing Linear Equations Page 17

Answer Key Page

3.

Glossary of Terminology, Formulas and Rules

Term Definition/Example Sum The answer to an addition problem.

Quotient The answer to a division problem.

Product The answer to a multiplication problem.

Difference The answer to a subtraction problem.

Absolute Value Make negative numbers positive and keep positive numbers positive.

Simplify To make less complex;

52

3

62

3

xx

x 2x2 is simplified

Solve for value Determine the value of the unknown; (x – 2)(x + 3) = 0 x = 2 and x = -3 are the solutions

Solve for a variable Isolate the desired variable, usually x or y (see Solving Equations on page 9)

Evaluate Determine the value when there is no unknown; 4(3 + 2) = 20 20 is the evaluated answer

Reciprocal Interchange the numerator and denominator. Given

4

5, the

reciprocal is 5

4. Given 3 (remember

33

1 ), the reciprocal is

1

3

Slope Rise over run or the change in y over the change in x

Distribute 3(x – 2) = 3x – 6

Coefficient The number in front of a variable

Variable A symbol that represents a number, usually a letter

Constant A number without a variable

Exponent The power to which a base is raised

Base The expression be raised to a power

Binomial The sum or difference of two terms

Trinomial The sum or difference of three terms

Polynomial The sum or difference of two or more terms

Like Terms Terms that have the same variables and powers

Greatest Common Factor (GCF) 9x3y2 – 6xy2 , 3xy2 is the GCF

Inequality Expressions that are not equal; 3 2 11x

Ordered Pair The x and y value of a point on a graph

Parabola The graph of a quadratic function

Proportion Statement of equal fractions

Domain The set of all possible x-values

Range The set of all possible y-values

Radical An expression that has a square root, cube root, etc.

Scatterplot A graph that contains a series of x and y coordinates

System of Equations Multiple equations containing multiple variables

4.

Solution The answer to an equation

Undefined Slope A vertical line has an undefined slope

Slope of Zero A horizontal line has a slope of zero

Perpendicular Two lines intersecting at a 90 angle

Parallel Two lines with the same slope

Formula Quadratic Formula

Given 2 0ax bx c ; 2 4

2

b b acx

a

Slope Given two coordinate pairs; 2 1

2 1

y ym

x x

Distance Given two coordinate pairs;

2 2

2 1 2 1d x x y y

Midpoint Given two coordinate pairs; 1 2 1 2,

2 2

x x y y

Pythagorean Theorem With legs a & b and hypotenuse c; 2 2 2a b c

Slope/Intercept Form of a Linear Equation

y = mx + b with m = slope, y-intercept at (0, b) and (x, y) represents any point on the line

5.

Factoring Polynomials

GCF stands for greatest common factor. This is the largest number (including variables) that divides evenly into a group of terms. Examples of finding the GCF of the following groups of terms.

1. First look to see whether they have a number factor in common. 2. Next, determine which variables each term has in common. Take

out the smallest exponent.

a. 4 7 2125 , 50 , 100x x x

b. 2 4 3 6, 4 , 40yx x y x

c.

7 2 2 33 , 18 , 6xy x y x y

25x2 x2y 3xy

In Algebra 1, you learn to multiply using the distributive property…. Example: Distribute… 4x(2x – 1) = 8x2 – 4x FACTORING is the opposite of multiplying. In other words, when you factor a polynomial, you write it as a multiplication problem. If a polynomial has a GCF, one way to factor it is to use the “reverse distributive property”.

Example: Factor… 23 27x x . What is the GCF of all of the terms? Write it on the line below, then work backwards with the distributive property to fill in the terms in the parenthesis.

___ ( x - 9) GCF

3 26 , 24 , 30x x x

1. 6, 24 & 30 all

have a 6 in common

2. x, x3, x2 all have

an x in common So the GCF is 6x

6.

Practice factoring the following polynomials:

1. 6 15x 2. 2 6x x 3. 39 3x x

TRINOMIALS When factoring a trinomial, you should always look for a GCF first. For now, we will focus on how to factor the trinomials.

Trinomials are usually of the form 2ax bx c . When a=1, the trinomial is very easy to factor…

Example: 2 9 20x x

(x - 4) (x - 5) (start with x times x to get 2x ) Try…

4. 2 5 6x x 5. 2 6 27x x 6. 2 4 21x x When a is a number greater than 1, the process requires more of a “guess and check” method.

Example: 22 21x x

(2x ___) (x ___) (Start with factors of 22x .)

(2x 3) (x 7) (Find factors of the last number and write them in. It’s okay to guess. You have to try something to get started. To see if you have the correct combination, test the inside and outside products. Can you get them to add up to the middle term? If not, switch your numbers around and try again.)

(2x 7) (x 3)

This combination works! The 7x would need to be negative, so the answer is (2x - 7) (x + 3).

3x

14x

What factors of 20 add up to -9?

These are not going to add up to -x, so try again.

7x

6x

Note: Diamond problems may help here!

7.

Try the following problems:

7. 25 22 8x x 8. 23 16 12a a 9. 24 11 3r r

MIXED FACTORING As stated earlier, you should always look for a GCF first before factoring the trinomial.

Example: What is the GCF of 215 6 21x x ? Factor this out first, then factor the trinomial. *3 is the GCF 3(5x2 – 2x – 7) 3(5x - 7)(x + 1) Try the following…

10. 26 2 8x x 11. 3 22 6x x x 12. 3 230 35 25x x x Factor the following polynomials completely:

13. 23 8 5n n 14. 29 4x 15. 2 10 24x x

16. 22 12x x 17. 2 9x 18. 3 23 6 18x x x

8.

SOLVING QUADRATIC EQUATIONS The key component to a quadratic equation is an equals sign. If it does not have an equals sign, it is not an equation and CANNOT be solved! At this point, we have learned two ways to solve quadratic equations: ZERO PRODUCT PROPERTY QUADRATIC FORMULA Must be factored & set equal Can be used to solve any to zero. quadratic equation. ZERO PRODUCT PROPERTY:

1. Set equation equal to zero. 2. Factor completely. 3. Set each factor equal to zero and solve.

Example: x2 – 4x = 21 1. x2 – 4x – 21= 0 2. (x – 7)(x +3) = 0 3. x – 7 = 0 and x + 3 = 0

x = 7 x = -3 Examples for you to try:

19. 2 6 16 0x x 20. 23 2 8x x 21. 3 222 40y y y

22. 2 8 10 0x x 23. 2 16 12x x 24. 27 10x x

9.

QUADRATIC FORMULA: 1. Set equation equal to zero. QUADRATIC FORMULA:

2. Apply quadratic formula. EXAMPLE:

1. 3x2 + 4x – 7 = 0

𝒙 =−𝟒 ± √(𝟒)𝟐 − (𝟒)(𝟑)(−𝟕)

𝟐(𝟑)

𝒙 =−𝟒 ± √𝟏𝟔 + 𝟖𝟒

𝟔

𝒙 =−𝟒 ± √𝟏𝟎𝟎

𝟔

𝒙 =−𝟒 ± 𝟏𝟎

𝟔

𝒙 = 𝟏,−𝟕

𝟐

Examples for you to try.

25. 23 10 4x x 26. 2 5 13x x 27. 22 5 10 0x x

ax2 +bx + c = 0

𝒙 =−𝒃 ± √𝒃𝟐 − 𝟒𝒂𝒄

𝟐𝒂

10.

SOLVING EQUATIONS FOR A VARIABLE SOLVING FOR Y: We often need to manipulate an equation so that it is solved for a different variable. For example, when graphing, it is often more helpful to have an equation in “y=” form. “Solve for y” does not necessarily mean that you will find a value for y. It means that the equation should say “y=” when you are finished. EXAMPLES: Solve the following equations for y.

7( 8) 3( 2) 8 7y x x

7y – 56 + 3x + 6 = 8x – 7 (Combine -56 + 6.) 7y – 50 + 3x = 8x – 7 (Subtract 3x from both sides.) 7y – 50 = 5x – 7 (Add 50 to each side.) 7y = 5x + 43 (Divide by 7 on both side.)

𝒚 =𝟓

𝟕𝒙 +

𝟒𝟑

𝟕

Try these examples:

28. 5 7

7x

y

29. 6 3 7 1 5 2x y x

𝟔( )𝒚−𝟐

Solving Systems of Linear Equations in Two Variables Solving by substitution Consider this system:

Look for the equation that is easiest to solve for x or y. In this case, we chose to solve the second equation for x. Now replace the x in the other equation with (-2 - 2y).

10 3 14

2 4 4

y x

x y

2 4 4

2 4 4

2 2

x y

x y

x y

10 3( 2 2 ) 14

10 6 6 14

16 6 14

16 8

0.5

y y

y y

y

y

y

11.

Find x by substituting 0.5 for y in either original equation. The solution is the coordinate pair (-3 , 0.5). NOTE: When both variables drop out of the equation, you either have “no solution” (false statement), or “infinitely many solutions” (true statement).

Solving by Elimination

First, rewrite the equation so that the x’s and y’s are lined up vertically. Next, decide what to multiply by to make the coefficients of either the x’s or the y’s the same numbers with opposite signs. Consider the same system as above: In this case we chose to make the x coefficients opposites: multiply the top equation by two and the bottom equation by three to get: If we add the two equations together, we eliminate the x: Finally, go back and substitute 0.5 for y in either original equation: Again, the solution must be written as a coordinate pair (-3 , 0.5). NOTE: When both variables drop out of the equation, you either have “no solution” (false statement), or “infinitely many solutions” (true statement).

Examples – Use either substitution or elimination. Determine which one would be easiest before beginning.

30. y = 2x - 3 31. 4x + 3y = 7

x + y = 15 2x – 9y = 35

2 2(.5)

3

x

x

10 3 14

4 2 4

y x

y x

20 6 28

12 6 12

y x

y x

32 16

0.5

y

y

10(0.5) 3 14

5 3 14

3 9

3

x

x

x

x

12.

Solve each of the following equations for y.

32. 8 5(3 2) 7 6x y x y

33. 3 1

4x

y

Solve the following systems of equations. Choose the best method for each problem. 34. x + y = -4 35. 3x – y = 1 -x + 2y = 13 y = 2x + 2 36. x = 4y +2 37. 2x + 3y = 10

x – 4y = 2 5x – 4y = 2 38. y = -4x + 3 39. 3x – 2y = -2 3x + 5y = -19 6x – 4y = 10

13.

EXPONENTS RULES: EXAMPLES: 1) xm ∙ xn = xm + n b5∙b2 =(bbbbb) ∙(bb)=b7 ; 29 ∙ 24 = 213

2) m

m n

n

xx

x

8

2

6

aa

a ;

546m

3m2m

3) m n m n(x ) x 4 10 40(x ) x ; 7 2 14(4b ) 16b

Example Problem: EXAMPLES for you to try:

40. 9 2

3 7

12

8

a b

a b 41.

72y

42. 2

3 43x y 43. 3 4x x

44. 2 3 9y v y v 45. 2 42 3xy x y

46. 5

2y 47. 3

25x y

48. 9 9

2 5

x y

x y 49.

75

10

x

x

11 5 2

20 6

2 11 2 5 2

20 6

22 10

20 6

22 20 10 6

2 4

(3 )

3

9

9

9

x z

x z

x z

x z

x z

x z

x z

x z

14.

RADICALS PERFECT SQUARES TO MEMORIZE: 12 = 1 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36 72 = 49 82 = 64 92 = 81 102 = 100 112 = 121 122 = 144

SIMPLIFYING RADICALS In order to simplify a square root, you must rewrite it as the multiplication of the factors! Be on the lookout for perfect squares.

For example, simplify the following square root: 45

What are factors of 45?

45 9 5

45 9 5 3 5

Ex 1: 27 Ex 2: 72

√9√3 = 3√3 √36√2 = 6√2 Simplify the following.

50. 75 51. 18 52. 48 53. 1000

SIMPLIFYING RADICALS – Rewrite the following radical expressions in simplest form.

54. 150 55. 24 56. 288

9 is the perfect square!

15.

Multiplying Polynomials You might want to use a generic rectangle. Examples:

a) (3 2)(4 5) x x b) 2( 1)( 2 3) x x x

3x - 2 x2 - 2x + 3 4x 12x2 -8x x x3 -2x2 3x + + 5 15x -10 1 x2 -2x 3 Multiply and simplify.

57. 2(2 3)x 58. (4 2)(3 5) x x 59. (2 3)( 7) x x

60. 2( 5)( 2 3)x x x

212 7 10x x 3 2 3x x x

16.

Writing the Equation of a Line Given Two Points Find the equation of the line through the given points. Example ( 6 , 5 ) and ( 9 , 2 ) Find the slope between the two lines using the slope formula. Plug this value into the general equation of a line, y = mx +b Also, pick either of the coordinate points to use and plug in the values of x and y into the equation. You can now solve the equation for b. Therefore the equation of the line is Find the equation of the line through the given points. 61. ( 0 , 4 ) and ( -1 , -5 ) 62. ( 4 , 2 ) and ( 8 , -1 )

2 1

2 1

2 5 31

9 6 3

y ym

x x

m

2 1(9)

2 9

11

b

b

b

1 11

11

y x

OR

y x

17.

GRAPHING LINEAR EQUATIONS When an equation is in slope – intercept form, it is easy to graph! Slope-Intercept Form: WARM UP: Graph the line y = ½ x + 3 Graph each of the following equations on separate sets of axes.

63) 2 7y x 64) 3

15

y x

Remember y = mx +b

-Start with b, the y-int: (0, 3)

-Use the slope m = ½ to graph the line

(rise over run, from the y-intercept go up

one, then right two)

18.

Solve the following for y and then graph. 65) 4 2 6x y 66) 4 2 12x y

67) –x + 3y = 27 68) –3 x + ¼ y = 2

19.

Answers to the practice problems throughout the packet. 1.) 3(2x + 5) 2.) x(x + 6) 3.) 3x(3x2 – 1) 4.) (x – 2)(x – 3)

5.) (x – 9)(x + 3) 6.) (x – 3)(x + 7) 7.) (5x – 2)(x – 4) 8.) (3a + 2)(a – 6)

9.) (4r + 1)(r – 3) 10.) 2(3x – 4)(x + 1) 11.) 2x(x + 3)(x – 1) 12.) 5x(2x + 1)(3x – 5)

13.) (3n + 5)(n + 1) 14.) (3x – 2)(3x + 2) 15.) (x – 12)(x + 2) 16.) 2x(x + 6)

17.) (x – 3)(x + 3) 18.) 3x(x2 – 2x + 6) 19.) x = -8, 2 20.)

4, 2

3x

21.) y = -20, -2, 0

22.) 4 6 x 23.) 6 2 5 x 24.) x = 2, 5

25.)

2 342.61,1.28

3

x

26.)

5 776.89, 1.89

2

x x

27.)

5 1053.81, 1.31

4

x

28.) 𝑦 =5

7𝑥 − 1

29.) 𝑦 = −1

7𝑥 +

9

7 30.) (6, 9) 31.) (4, -3)

32.) 10

9

xy

33.) 3 1

4

xy

34.) (-7, 3) 35.) (3, 8) 36.) Infinitely many solutions

37.) (2, 2) 38.) (2, -5) 39.) No Solution 40.)

6

5

3

2

a

b

41.) 7128y 42.) 6 89x y 43.) x7 44.) y5v10

45.) 6x5y3 46.) y10 47.) 6 3125x y 48.) 7 4x y

49.) 6

2

x

50.) 5 3 51.) 3 2 52.) 4 3

53.) 10 10 54.) 5 6 55.) 2 6 56.) 12 2

57.) 4x2 - 12x + 9 58.) 12x2 +14x – 10 59.) 2x2 – 11x – 21 60.) x3 – 3x2 - 13x +15

61.) y = 9x +4 62.) 𝑦 = −3

4𝑥 + 5