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CST Prep Part II MAIN MENU. Designed by Ms.Carranza and Mrs. Murray Solved by: 8 th Grade Gate Students 2011. Standard 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multi-step problems, including word problems, by using these techniques. - PowerPoint PPT Presentation
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CST Prep Part IIMAIN MENU
Standard 10
Standard 11
Standard 12
Standard 13
Standard 14
Standard 15
Standard 16
Standard 17
Standard 18
Standard 19
Standard 20
Standard 21
Standard 22
Standard 23
Standard 25.1
Designed by Ms.Carranza and Mrs. Murray Solved by: 8th Grade Gate Students 2011
Standard 10.0 Students add, subtract, multiply, and divide monomials and polynomials.
Students solve multi-step problems, including word problems, by using these techniques.
Problem 47 Problem 49
Problem 51
Problem 48
Problem 50Main Menu
Problem 52
51) A volleyball court is shaped like a rectangle. It has a width of x meters and length of 2x meters. Which
expression gives the area of the court in the square meters?
A 3xB 2x²C 3x²D 2x³
Vocabulary Rules & Strategies
Solution & Answer Standard 10
Vocabulary
• Expression : a mathematical phrase that contains operations, numbers, and/or variables.
• Area: The number of non overlapping unit squares of a given size that will exactly cover the interior of a plane figure.
• Square Meters: a unit of area measurement equal to a square measuring one meter on each side.
Back to Problem
Rules & Strategies
• Area= base (length) * height (width)• Product of Powers: combine base, add
powers• Add exponents ( x = x )
Back to Problem
1
x
2x
Solution
• 1x
2x
Answer: BBack to Problem
Standard 10
2x
2
1 1
50) Which of the following expressions
is equal to (x+2)+(x-2)(2x+1)?
A. 2x² - 2x
B. 2x² - 4x
C. 2x² + x
D. 4x² + 2x
Rules and Strategies
Solution and Answer
Standard 10
Vocabulary
Solution and Answers (x+2)+(x-2x)(2x+1)?
Step 1: Area Method
Step 2: Combine like terms and put in Descending order
Answer: 2x² - 2x
2x + 12x + 1 xx -2 2x²-3x-2-2 2x²-3x-2
1X + 2 + 2x² - 3x - 21X + 2 + 2x² - 3x - 2
2x²2x² +x+x
-4x-4x -2-2
-2x-2x2x²2x²0+0+
Back to Problem
Standard 10
Vocabulary
Area Method: Area Method: ax + bax + b
(ax + b)(ax + b) (ax + b)(ax + b) axax b b aax aax + + (abx + abx) (abx + abx) + + bbbb
Descending Order: Descending Order: ordering terms from greatest to least((ex. 1x+ 2x² + 3x+1x³ + 9 = 1x³ + 2x² + 3x + 1x + 9)ex. 1x+ 2x² + 3x+1x³ + 9 = 1x³ + 2x² + 3x + 1x + 9)
abxabx
aaxaax abxabx
bbbb
Back to Problem
Rules & Strategies
11st.st. Use the Area Method22nd.nd. Compare like terms33rd.rd. Descending Order
Back to Problem
49)The sum of the two binomials is 5x2-6x, If one of the binomials is 3x2-2x, what is the
other binomial
A 2x2-4xB 2x2-8xC 8x2+4xD 8x2-8x
Vocabulary Rules & Strategies
Standard 10Solution and
Answer
Vocabulary
• sum– answer t an addition problem • Binomial- A polynomial with two terms
Back to Problem
Rules and strategies
• Line up the binomials• Combine both binomials by subtracting
Back to Problem
Solution
(3x2-2x)+2x2-4x5x2-6x
Answer: A Standard 10
1st binomial
2nd binomial
Sum of binomials
Back to Problem
47) 5x3
10x7
A) 2x4
B)1 2x4
C) 1 5x4
D) X4
5
Vocabulary Rules & Strategies
Standard 10.0Solution and
AnswerSolution and
Answer
Vocabulary
• Quotient of Powers: simplify coefficientscombine base
subtract exponents
Back to Problem
Rules & Strategies
• In order to skip the negative exponent rule, circle the biggest exponent to tell if the variable stays in the denominator or numerator.
• Subtract powers • Divide if there is any whole numbers in the
numerator and denominator.
Back to Problem
Solution
5x3 1
10x7 2x4
Answer: B Back to Problem Standard 10.0
48.) (4x²-2x+8) – (x²+3x-2)
Vocabulary Rules and Strategies
Solution and Answer Standard 10
Vocabulary
• Parenthesis ( ); indicates separate grouping of symbols
• Exponent ²; a symbol or number placed above and after another symbol or number to denote the power to which the latter is to be raised
• Variable ‘x’; a quantity or function that may assume any given value or set of values
Back to Problem
Rules and Strategies
• Subtracting Polynomials / Lesson 7-7• Distribute (-) before grouping
Back to Problem
Solution and Answer
Answer:D Back to Problem Standard 10
(4x²-2x+8) – 1 (x²+3x-2)(4x²-2x+8)-x²-3x+2
(4x²-1x²) + (-2x-3x) + (8+2)
3x²-5x+10
Standard 11 Students solve multistep problems involving linear equations and
inequalities in one variable.
Problem 54
Problem 56
Problem 53
Problem 55
Main Menu
53)Which is the factored form of 3a²-24ab+48b²?
a. (3a-8b)(a-6b)b. (3a-16b)(a-3b)c. 3(a-4b)(a-4b)d. 3(a-8b)(a-8b)
Vocabulary Rules & Strategies
Solution & Answer Standard 11
Vocabulary
• Factored form= Form of equation in which each term is simplified from factoring methods and GCF.
Back to Problem
Rules & Strategies
• Ask yourself if you can factor out a GCF• Use the diamond method• Keep asking yourself whether you can factor more.If there is a GCF, don’t forget to Include it in your final answer.
Back to Problem
Solution & Answer
Factor out a gcf3a²-24ab + 48b²
3 3 3 3(a²-8ab+16b²) +16
keep factoring!! a a -4b -4b
3 (a-4b)2
Solution: C -8 Back to Problem
Standard 11
54) Which is the factor of x² – 11x + 24
A x + 3 B x - 3
C x + 4 D x - 4
Vocabulary Rules & Strategies
Solution & Answer Standard 11
Vocabulary
• Factor = A number or expression that is multiplied by another number or expression to get a product.
• Term = a part of an expression that is added or subtracted
• Binomial = 2 terms• Trinomial = 3 terms
Back to Problem
Rules & Strategies• Since there are three terms in this trinomial you use the “Diamond Method”• Diamond Method Formula : ax² + bx + c• Make an “X” on your paper.• On the top intersect write a multiplication sign (x) to show that the two
denominators multiply to equal the number you're going to get above the multiplication sign.
• On the bottom intersect write a plus sign (+) to show that the two denominators added together equal the number you’re going to get below the addition sign.
• Plug the value for “c” above the multiplication sign ( +24 )• Plug in the value for “b” below the addition sign ( -11 )• Now , think of two numbers that multiplied equal +24 and added equal -11 . • ( You should come up with -3 and -8 )• Lastly , you rewrite your fractions as binomials : 1x = ( x – 3 ) -3
Back to Problem
Solution
x² – 11x + 24 + 24 1x x 1x -3 + -8 - 11 ( x – 3 ) ( x – 8 ) Final answer
Answer: BBack to Problem Standard 11
55) Which of the following shows 9t + 12t + 4 factored completely?
A (3t + 2) B (3t + 4) (3t +1) C (9t + 4) (t + 1)
D 9t + 12t + 4
Vocabulary Rules & Strategies
Solution & Answer
2
2
2
Standard 11
Vocabulary
• Factored :one of two or more numbers, algebraic expressions, or the like, that when multiplied together produce a given product; a divisor.
Back to Problem
Rules & Strategies
• Always ask yourself which factoring method should I use?
• You should always see if you can factor out a GCF
• Use“diamond method” when there is no GCF and the coefficient is greater than one.
Back to Problem
Solution
9t + 12t + 4
+36 3 9t 9t 3 3t 3 +6 +6 3 2
+12
(3t+2) Back to
Problem
2
3t 2
•Make sure to simplify the fractions
2Answer: A
Standard 11
56) What is the complete factorization of 32-8z²
A. -8(2+z)(2-z)B. 8(2+z)(2-z)
C. -8(2+z)²D. 8(2-z)²
Vocabulary Rules & Strategies
Standard 11Solution &
Answer
Vocabulary
• Factorization: to simplify
Back to Problem
Rules & Strategies
• Find GCF• Divide each term by GCF• Rewrite equation w/ GCF outside of parenthesis• Divide inside terms by a negative • Rewrite the equation w/ negative on the outside• Difference of 2 Squares• Rewrite new equation & THAT’S YOUR ANSWER !
Back to Problem
Solution
32-8z²-8 -8
factor out a -8 to get a positive z²-8(z²-4) keep factoring diff of 2 squares
-8(z+2)(z-2)Or
-8(2+z)(2-z)
Answer: A. -8(2+z)(2-z) Back to Problem
Standard 11
Standard 12
• Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.
Problem 78 Problem 79 Problem 80
Problem 81Main Menu
Problem 77
78.) 6x2 + 21x + 9 4x2 - 1
Vocabulary Rules and Strategies
Solution and Answer Standard 12
Vocabulary
• GCF ; Greatest Common Factor • Super Diamond ; x2 + bx + x• Difference of Two Squares ; terms need to be in
perfect squares [Example: (a+b)(a-b)]
Back to Problem
Rules and Strategies
• Top ; find GCF, super diamond• Bottom ; difference of two squares, terms
need to be perfect squares
Back to Problem
Solution and Answer
Back to Problem Standard 12
6x2 + 21x + 9
Top/Numerator ; 6x2: 2 ∙ 3 ∙ x ∙ x21x: 3 ∙ 7 ∙ x 9: 3 ∙ 3
GCF: 3
6x 2 +21 + 9 3 3 33 ( 2x 2 + 7x + 3)
Super Diamond!!
1) GCF2) Difference of Two Squares3)Cross Cancel
∙+
+6
+7
+2x+6
+2x+1
x
+3
3(x+3)(2x+1)
Bottom/Denominator ;4x2 - 1
2x 2x 1 1(2x+1)(2x-1)
Top/Numerator ;
6x2 + 21x + 9 4x2 – 1
Cross Cancel3(x+3)(2x+1)(2x+1)(2x-1)
=Answer ; B 3(x+3) 2x-1
=
79)What is x2-4x+4 reduced to lowest terms?
x2-3x+2A) x-2 X-1B) x-2 X+1C) x+2 X-1D) x+2 X+1
Vocabulary Rules & Strategies
Solution & Answer Standard 12
Vocabulary
• Reduce = lower in degree• Lowest term = the form of a fraction after
dividing the numerator and denominator by their greatest common divisor.
Back to Problem
Rules & Strategies
• First you should ask yourself which strategy is best to change the equation into a dividable state using either; Greatest common facture (GCF), difference of 2 squares, or one of the diamond methods.
• In this case the best one would be the diamond method on both the denominator and numerator.
• Once you find the factors of both of the trinomials reduce by dividing the like terms.
Back to Problem
Solution ax2+bx+c=0
Step 1: x2-4x+4 x2-3x+2
Step 2: x2-4x+4 (x-2) (x-2) x2-3x+2 (x-2) (x-1) Diamond Method
4 2 x * x x * x -2 -2 -2 -1 + +
-4 Divide out! -3
(x-2) (x-2) = (x-2) Answer: A (x-2) (x-1) (x-1) Back to
Problem Standard 12
80. What is 12a3 – 20a2 reduced to lowest terms? 16a2 + 8a
Vocabulary Rules and Strategies
Solution and Answer Standard 12
Vocabulary
• Reduced ; simplified or lowest terms
Back to Problem
Rules and Strategies
• Find GCF for both numerator and denominator
• Simplify to lowest terms
Back to Problem
Solution and Answer 12a3 - 20a2 GCF:4a2 16a2 + 8a GCF: 8a2
4 a2 (3a – 5) *Simplify 8 a (2a + 1)
a (3a – 5)2 (2a +1)
Answer:D Back to Problem Standard 12
Numerator :12a3 - 20a2
4a2 4a2
4a2 (3a – 5)
Denominator:16a2 + 8a 8a 8a8a (2a + 1)
81) What is the simplest for of the fraction: _x2-1_ x2+x-2
A)_-1__ x-2B)x-1 x-2C)x-1 x+2D)x+1 x+2
Vocabulary Rules & Strategies
Solution & Answer Standard 12
Vocabulary
• Simplest form of a rational expression- A rational expression is in simplest form if the numerator and denominator have no common factors. Ex:
x2-1 (x-1)(x+1)x2+x-2 (x+1)(x+2) (x+1) (x+2) Simplest Form Back to
Problem
Rules & Strategies
• First ask yourself: “1)How will I factor?2)Simplify/ Divide out
*make sure to separate work to make your work easier to read.
Back to Problem
1.GCF2.Diamond
3.SuperDiamond4.Difference of 2 Squares
Solution
Answer:DBack to Problem Standard 12
_x2-1_ x2+x-2
WORK:Step 1) difference of 2 squares/diamond
Difference of 2 squares:
x2- 1x x 1 1 (x-1)(x+1) Diamond:
x x-1 +2
+1
-2
(x-1)(x+2)
(x-1)(x+1)(x-1)(x+2)
Standard 13 Students add, subtract, multiply, and divide rational expressions
and functions. Students solve both computationally and conceptually challenging problems by using these techniques.
Problem 82 Problem 84Problem 83
Problem 85 Main Menu
82) 7z²+7z • z²-4 4z+8 z³+2z²+z
a. 7(z-2) b.7(z+2) 4(z+1) 4(z-1)c. 7z(z+1) d. 7z(z-1) 4(z+2) 4(z+2)
Vocabulary Rules & Strategies
Solution & Answer
Standard 13
Vocabulary
Factoring= The process of writing a number or algebraic expression as a product.
GCF= For two or more numbers, the largest whole number that divides evenly into each number.
Back to Problem
Rules & Strategies
Check whether to use GCF, before choosing a factoring method.
Divide out common binomials.
Back to Problem
Solution & AnswerGCF:7z -> 7z²+7z • z²-4 <- Difference of 2 SquaresGCF: 4 -> 4z+8 z³+2z²+z <- GCF: z; Diamond
7z(z+1) • (z+2)(z-2) 4(z+2) z (z+1) (z+1)
7z(z+1) • (z-2) 4 z (z+1) (z+1)
Answer: A = 7(z-2) 4(z+1) Back to
Problem
Standard 13
84) x+8x+16 2x+8 x+3 x-9
A 2(x+4) (x-3)(x+3)B 2(x+3)(x-3) x+4
Vocabulary Rules & Strategies
Solution & Answer part 1 Standard 13
2
2
C (x+4)(x-3) 2D (x+4)(x-3) 2(x+3)
2
22
Vocabulary
Rational Expression: a quotient of polynomials
Back to Problem
Rules & Strategies
• Take the reciprocal of the fraction on the right side of the sign and then multiply.
• Factor each term if needed.• Divide out common factors, if needed.
Back to Problem
Solutionx+8x+16 2x+8 x+8x+16 x-9 x+3 x-9 x+3 2x+8x+8x+16 2x+8 x-9 2x: 2 x x x 3 3 (x+4)(x+4) (x+3)(x-3)
8: 2 2 2 (x+3) (x-3) x+3 2(x+4)
Diivide out!
(x+4) (x-3)
2 answer: C
Back to
Problem Standard 13
2
2
2 2
Greatest Common Factor: 2
Difference of 2
+16
+8
+
.+4 +41x 1x
(x+4)
GCF: 2
2x+822
2(x+4)
2
Change to multiplication and take reciprocal of second rational
expression
Change to multiplication and take reciprocal of second rational
expression
LOOK BEFORE YOU BEGIN!
2
W
R
O
K
85) Which fraction is equivalent to 3x5
x x4 + 2
A. X² 5B. 9x² 20
c. 4 5D. 9 5
Vocabulary Rules &
Strategies
Solution & Answer Standard 13
Vocabulary
• Fraction: a ratio of two expressions or numbers other than zero
• Equivalent: equal
Back to Problem
Rules & Strategies
• Find the least common denominator (LCD)• Change division into multiplication• With the second fraction, take the reciprocal• Cross cancel• Multiply straight across
Back to Problem
Solution & Answer3x5
x x4 + 2
x x 4 = 4
x 2x+ 2 = 4
3x take reciprocal
4Answer: C. 3x 4 4 5 3x 5 Back to
Problem
Standard 13
find Common denominator = 4
find Common denominator = 4
83) Which fraction equals the product?
A 2x -3 3x+2
B 3x+2 4x-3
C x²-25 6x²-5x-6
D 2x²+7x-15 3X²-13X-10
Vocabulary Rules & Strategies
Solution & Answer Standard 13
Vocabulary
• Fraction = a ratio of algebraic quantities similarly expressed
• Product = the answer of a multiplication problem • Area Method = multiply outside add or subtract
inside terms (like terms)• Numerator = the top numbers of a fraction• Denominator = the bottom numbers of a fractionBinomials: two terms Back to
Problem
Rules & Strategies• Since there are two different fractions in parenthesis and both the numerators and
denominators are binomials you use “Area Method”• Area method is when you draw out a rectangle and divide it into 4 quadrants (4 squares)• Then, you put one binomial on the left side written out like this:
• Next, you put the other binomial adjacent (next to) the fraction on top of the rectangle • Then you multiply each term to each other and Place each number in a square (label them 1, 2, 3, and 4)• Write out numbers 1 and 4 into an expression as the first andlast terms : 2x² + 15• Now, since numbers 2 and 3 are like terms combine them and putthem in between 2x² and +15 : 2x² +7x +15• Repeat this process to the other binomial and place the expression underneath 2x² +7x +15 .
Back to Problem
Solution
Answer: D Back to Problem Standard 13
3x²-13x-10
+5
+2x -3
2x²
10x
-3x
-15
²
-5+x
+3x
+2
3x² -15x
2x -10
2x²+7x-15 Combine2x²+7x-15 on the top and 3x²-13-10on the bottom
+x
Standard 14 Students solve a quadratic equation by factoring or completing
the square.
Problem 57 Problem 59
Problem 61
Problem 58
Problem 62
Problem 60 Main Menu
59) What are the solutions for the quadratic equation x + 6x = 16?
A -2,-8B -2, 8C 2,-8D 2,8
Vocabulary Rules & Strategies
Solution & Answer Standard 14
2
Vocabulary
• Quadratic equation : an equation containing a single variable of degree 2.
Back to Problem
Rules & Strategies
• the “diamond method”• All the numbers have to be on one side of the
equation.
Back to Problem
Solution
ax + bx + c = 0x +6x = 16
-16 -16 x + 6x – 16 =0
(x + 8) (x - 2) = 0 Solve each factor to = 0
x + 8 = 0 x – 2 = 0 - 8 - 8 +2 +2 x = -8 x = 2
Back to Problem Standard 14
-16
+6
x-2
x+8
Answer: C
2
2
2
57) If x is added to x, the sum is 42. Which of the following could be the
value of x?
A -7B -6C 14D 42
Vocabulary Rules & Strategies
Solution & Answer Standard 14
2
Vocabulary
• Value: A number represented by a figure, symbol; the value of x.
• Sum: a series of numbers to be added up
Back to Problem
Rules & Strategies
• Plug in the value of x• Set up the expression (x) + x
Back to Problem
2
Solution
Try x=-7 (x) + x= 42
(-7) +(-7)= 42 49 + (-7)=42
42=42 Answer: A Back to
Problem Standard 14
2
2
62) What are the solutions for the quadratic equation x2-8x=9?
A) 3B) 3,-3C) 1,-9D) -1,9
Vocabulary Rules & Strategies
Solution & Answer Standard 14
Vocabulary
• Solutions - the answer itself• Quadratic - involving the square and no higher
power of the unknown quantity; of the second degree.
• Equation - an expression or a proposition, often algebraic, asserting the equality of two quantities.
Back to Problem
Rules & Strategies
• First get into Quadratic Form ax2+bx+c=0.• Then use the diamond method• Finally separate the two factors to = 0.• Solve for “X”
Back to Problem
Solution ax2+bx+c=0
Step 1: x2-8x=9 Step 3: (x-9) (x+1) = 0 Zero Product -9 -9 X-9=0 x+1=0 Property +9=9 -1=-1 Step 2: 1x2-8x-9=0 x=9 x=-1 Answers Diamond Method
-9 x * x +1 -9 +
-8
Answer: DBack to Problem Standard 14
58) What quantity should be added to both sides of this equation to complete
the square?x²-8x=5
A. 4B. -4C. 16D. -16
Vocabulary Rules & Strategies
Solution Standard 14
Vocabulary
• Quantity: amount
Back to Problem
Rules & Strategies
• Identify “b”• Plug in “b” to the equation: b ²
2• Solve to complete the square.• The solution to your equation is the answer
Back to Problem
Solution
b=-8 -8 ²
2(-4)²+16
Answer: C. 16Back to Problem
Standard 14
60) Leanne correctly solved the equation x² + 4x = 6 by completing the square. Which equation is part of her solution?
A ( x + 2 )² = 8 B ( x + 2)² = 10 C ( x + 4 )² = 10 D ( x + 4)² = 22
Vocabulary Rules & Strategies
Solution & Answer Standard 14
Vocabulary• Completing the square : a method for solving
quadratic equations by using steps such as1. Making sure you have the formula x² + bx + c 2. Find b ² 23. Completing the square with the answer to by adding it to both sides4. Factor the left and simplify the right 5. Use Square root property then you’re done Back to
Problem
Rules & Strategies
• Use the factoring method: completing the square
Back to Problem
Solution
x² + 4x = 6
= b = 4 = 2 ² = 4 add to both sides of the equation
2 2 Perfect Square Trinomial : X² + 4x + 4 = 6 + 4 factor the right, simplify the left factor by diamond methodFinal Answer : ( x + 2 )² = 10 take square root of both sides
Answer: BBack to Problem Standard 14
Completing the Square
Completing the Square
61) Carter is solving this equation by factoring.
10x²-25x+15=0Which expression could be one of his correct
factors?a. x+3b.x-3
c.2x+3d. 2x-3
Vocabulary Rules & Strategies
Solution & Answer
Standard 14
Vocabulary
Expression does not have an = Factors can also be the binomials you get when you diamond
Back to Problem
Rules & Strategies
• Ask yourself which factoring method do I use?• GCF? Diamond? Super Diamond? Diff of 2
Squares• Make sure to find two terms which can be
multiplied to the top, but added to the bottom of the diamond.
• Keep asking yourself whether to factor more or not.
Back to Problem
Solution & AnswerGCF: 5
10x²-25x+15=0 5 5 5 5
5 ( 2x² -5x +3) =0 +6 5( x-1) (2x-3)=0 x = 2x 2x -1 -2 -5 -3 These are all factors, but
they are just asking for one of them.Answer: D Back to
Problem Standard 14
Standard 15 Students apply algebraic techniques to solve rate problems, work
problems, and percent mixture problems.
Problem 85
Problem 91
Problem 87
Problem 89
Problem 86
Problem 90
Problem 88 Main Menu
87) Andy’s average driving speed for a 4-hour trip was 45 miles per hour. During the first 3 hours he drove 40 miles per
hour. What was his average speed for the last hour of his trip?
A 50 miles per hourB 60 miles per hourC 65 miles per hourD 70 miles per hour
Vocabulary Rules & Strategies
Solution& Answer Standard 15
Vocabulary
• Rate = A ratio that compares two quantities measured in different units.
• Average = The sum of the values in a data set divided by the number of data values. Also called the mean.
Back to Problem
Rules & Strategies
• Set up formula distance = rate time• Put the same units together
Back to Problem
.
Solutiond = 45 4d = 180
180 = r 3 isolate rate 3 360 = r
Answer: B Back to Problem Standard 15
.
.
86) A pharmacist mixed some 10%-saline solution with some 15%- saline solution to obtain 100mL of a
12%-saline solution. How much of the 10%-saline solution did the pharmacist use in the mixture?
A) 60mLB) 45mLC) 40mL
D) 25mL
Solution & Answer
Solution & Answer
VocabularyVocabularyRules &
StrategiesRules &
Strategies
Standard 15
Standard 15
VocabularyDistributive PropertyDistributive Property: is when you
distribute what is outside of the parentheses.
Isolate the variable: Isolate the variable: is to simplify the equation using operations to get the
variable alone , on one side of the equal sign.
Back to ProblemBack to Problem
x + y Solution
=100mL
x+y= 100 isolate y -> y=(100-x) Work 0.10x+0.15y=12 -0.15x0.10x+ 0.15(100-x)=12 -0.05x=-3 +0.10x 0.10x+15-0.15x=12 -0.05 -0.05 -0.05x -15 -15
0.10x-0.15x= -3 Answer: A
-0.05x=-3 isolate xBack to
ProblemBack to
Problem
10
15
12%
10% + 15% = 12%
x + y = 100mL
.10x + .15y = .12(100)=12
X=60mLX=60mL
Standard 15
100 mL
Rules & Strategies
• Make a drawing with the values
• Remember that you need to go two spaces to the left when changing a value to a decimal
• Make a system of equation and use substitution
Back to ProblemBack to Problem
88) One pipe can fill a tank in 20 minutes. While another pipe takes
30 minutes to fill the same tank. How long would it take the two pipes together to fill the tank ?
a. 50minb. 25minc. 15mind. 12min
Vocabulary
Solution and Answer
Rules and Strategies
Standard 15
Vocabulary
• None available
Back to Problem
Rules & StrategiesFind the number of minutes it takes to fill the tank
together. There are 60 minutes in 1 hourPut information into fraction (over 1) and find the LCM.
Once LCM is found, distribute the LCM inside whatever is in the parentheses .
Isolate “m” and your answer will be found.
Back to Problem
Solution 1 1 20m + 30m = 1 m=minutes
LCM: 20: 22 5 Choose one with the greatest power 30: 2 5 3LCM: 22 . 5 . 3 = 60 Distribute 60 to original equation 60 1 1 20m + 30m = 1(60) 60 60 20m + 30m= 60 Reduce Fractions 3m + 2m= 60 5m= 60 5 5
Answer: D m=12
Back to Problem Standard 15
89) Two airplanes left the same airport traveling in opposite directions. If one airplane averages 400 miles per hour and
the other airplane averages 250 miles per hour, in how many hours will the distance between the two planes be 1625
miles?
A 2.5B 4C 5
D 10.8
Vocabulary Rules & Strategies
Solution & Answer Standard 15
Vocabulary
• Distance= rate * time isolate time • d= rt r r t= d r
Back to Problem
Rules & Strategies
• Remember to add both planes’ averages • Divide answer to total distance
Back to Problem
Solution 1st plane 2nd plane
avg 400 avg 250 400 + 250 = 650 is the rate time = 1625 (distance)
650(rate) time= 2.5 hours
Back to Problem Standard 15
90) Lisa will make punch that is 25% fruit juice by adding pure fruit juice to a 2-liter mixture that is 10% pure fruit
juice. How many liters of pure fruit juice does she need to add?
A. 0.4 literB. 0.5 literC. 2 litersD. 8 liters
Vocabulary Rules & Strategies
Solution Standard 15
Vocabulary
• Pure=100% or .10• Liters=amount
Back to Problem
Rules & Strategies
• Use substitution• Goal: solve for y• First, isolate “x” then plug it in the 2nd
equation• NOTE: 2 liters of 10% is 2(.10)= 0.2
Back to Problem
Solution and Answerx= 25 % = .25 y= 100% because its it's PURE fruit juice = 1 Solve for y
Results in 2 liters of 10% = (.10) fruit juiceFirst, isolate x then plug it in.x-y = 2 .25x - 1y = 2(.10)
x-y=2-y=2-x (2.4) – y = 2-1 -1 -1 -2.4 -2.4y= -2+x -y = -.4
-1 -1.25x-(-2+x)=.2.25x+2-x=.2 y= .4 -2 -2.25x-x= -1.8
-.75x = -1.8-.75= -.75 x = 2.4
Back to Problem
Standard 15
91) Jenna's car averaged 30 miles per gallon of gasoline on her trip. What is the value of x in gallons of gasoline?
A 32B 41 C 55 D 80
Vocabulary Rules & Strategies
Solution & Answer Standard 15
Miles traveled
600 450 300 960
Gallons of gasoline
20 15 10 x
Vocabulary• Average = The sum of all the values in a data set
divided by the number of data values. Also called the mean.
• Value = The amount of ; or equivalency of .• Rate = A ratio that compares two quantities
measured in different units.• Ratio = A comparison of two numbers of
quantities .• Equation = A mathematical statement that two
expressions equal.Back to Problem
Rules & Strategies
• Rewrite the problem as an equation and solve for ‘’X’’ ( 960 = 30x ) * The ration is 960 miles = x number of gallons and you put the “30” in because its miles per gallon.*
• When you have the equation “960 = 30x” divide 30 on both sides to isolate “x”.
Back to Problem
Solution
960 = 30x 30 30 x = 32
Answer: A
Back to Problem Standard 15
Standard 1616. Students understand the concepts of a relation and a function, determine
whether a given relation defines a function, and give pertinent information about given relations and functions.
Problem 92
Problem 93
Main Menu
92) Beth is two years older than Julio. Gerald is twice as old as Beth.
Debra is twice as old as Gerald. The sum of their ages is 38. How old is Beth?
A 3B 5C 6D 8
Vocabulary Rules& Strategies
Solution & Answer Standard 16
Vocabulary
• Twice - Double• Sum - Answer to an addition problem• Two years older - x +2
Back to Problem
Rules & Strategies
Given: Find: Beth’s age• Beth is x• Julio is x-2• Gerald is 2x• Debra is 2(2x)• Total years of age is 38
Back to Problem
Add all of them to make the expression equivalent to 38
Solution
x+(x-2) + 2x + 2(2x) = 38x + x - 2 + 2x + 4x = 38 +2 +28x = 408x = 408 8x=5
Answer: B Back to Problem Standard 16
Beth is five years old.
Combine like terms
Vocabulary Rules & Strategies
Solution & Answer Standard 16
A.Input Output
1 2
2 2
3 3
4 3
B.Input Output
2 6
2 5
6 4
6 3
C.Input Output
1 2
2 4
4 6
4 8
D.Input Output
0 1
0 2
1 3
1 4
93) Which relation is a function?
Vocabulary
• Relation = a property that associates two quantities in a definite order, as equality or inequality
• Function = set of ordered pairs in which none of the first elements of the pairs appears twice
Back to Problem
Rules & Strategies
• Look @ input (x values) and the numbers in the x value can not be repeated twice.
Back to Problem
Solution
Answer: A Back to Problem Standard 16
Input Output
1 2
2 2
3 3
4 3
Standard 17 Students determine the domain of independent variables and the range of dependent variables
defined by a graph, a set of ordered pairs, or a symbolic expression.
Problem 94 Problem 95 Main Menu
94) For which equation graphed below are all the y-values negative?
Vocabulary Rules and Strategies
Solution and Answer
Standard 17
a. b. c. d.
Vocabulary
• y-value ; (x, y)
Back to Problem
Rules and Strategies
• Lines must cross at negative y-int. (Quadrant 2 & 3) in order for all to be negative.
Back to Problem
Solution and Answer
Lines crossin the y-intof -1.
Answer: AStandard 17 Back to
Problem
14
23
95)What is the domain of the function shown on the graph below?
A {-1,-2,-3,-4}B {-1,-2,-4,-5}C {1,2,3,4}D {1,2,4,5}
Vocabulary Rules & Strategies
Solution & Answer Standard 17
Vocabulary
• Domain=the set of all first coordinates (x-values) of a relation or function
• Function= A relation in which every domain value is paired with exactly one range value
Back to Problem
Rules & Strategies
• Identify the ordered pairs (x,y)• Put them in a t –chart• Remember domain= x-values
Back to Problem
Solution
Back to Problem Standard 17Answer: D; {1,2,4,5}
X y1 -1
2 -2
4 -4
5 -5
*Only focus on domain
Standard 1818. Students determine whether a relation defined by a graph, a set of
ordered pairs, or symbolic expression is a function and justify the conclusion.
Problem 96Problem 96Problem 96Problem 96 Main Menu
96) 96) Which of the following graphs represents a relation that is not a function of x?
Click here to look at graphs.VocabularyVocabularyVocabularyVocabulary
Solution and Solution and answeranswer
Solution and Solution and answeranswer
Rules and Rules and SrategiesSrategiesRules and Rules and SrategiesSrategies
Standard 18 Standard 18 Standard 18 Standard 18
A A CC
BB D D
Back to Back to problemproblem Back to Back to
problemproblem
vocabularyvocabulary1.1. Function_Function_ : A relation ship or expression
involving one or more variables. (y=mx+b)
2.2. Vertical line test: Vertical line test: putting strait lines going through X Axis.
It tests to see
Whether it is a
function or
not. Back to Back to problemproblem Back to Back to
problemproblem
Solution Solution and and
answeranswerRemember
To Give each
a vertical line test D, isn`t a function Because it is being
hit more than once.
Back to Back to problemproblem Back to Back to problemproblem
Standard Standard 1818
Standard Standard 1818
Rules and Strategies Rules and Strategies You have to give it the vertical line test in order to see if it is a function.
Rule: it is only a function if it only hits the line once.
Back to Back to problemproblem Back to Back to
problemproblem
Standard 19 Students know the quadratic formula and are familiar with its
proof by completing the square.
Problem 64
Problem 63
Main Menu
64) Four steps to derive the quadratic formula are shown below. What is the correct order for these
steps?
A) I, IV, II, IIIB) I, III, IV, IIC) II, IV, I, IIID) II, III, I, IV
Vocabulary Rules & Strategies
Solution & Answer Standard 19
Vocabulary
Derive= to reach or obtain by reasoning; deduce; infer. Quadratic Formula= the formula for determining the
roots of a quadratic equation from its coefficients.Quadratic= involving the square and no higher power
of the unknown quantity; of the second degree. Formula= a general relationship, principle, or rule
stated, often as an equation, in the form of symbols.
Back to Problem
Rules & Strategies Quadratic Proof
This is the
order a quadratic
formula proof should look like.
Back to Problem
Solution
Solution: A
Back to Problem Standard 19
#63Toni is solving this equation by completing the square.Step1: ax2 + bx =-cStep 2: x2 + b
a X= - ca
Step 3:?
Which should be step 3 in the solution?
A)X2 =-cb - b
aX B)X +ba = - c
ax
C)X 2 + baX+b
2a = -ca + b
2a D)X 2 +baX+ b
2a 2 = - c
a + b2a
2
VocabularyVocabulary Rules and SrategiesRules and Srategies
Solution and answer
Solution and answer Standard 19 Standard 19
Vocabulary1.1. Completing the square: Completing the square: An expression in the
form of x 2 + bx is not a perfect square. However, you can use the relationship above to add a term to x 2 + bx to form a trinomial that can be a perfect square.
Back to problemBack to problem
Solution and AnswerStep1: ax2 + bx =-c b/a 2Step 2: x2 + b
a X= - ca 2
Step 3:? ba x 1
2 2= b2a 2
x2 + ba X+ b
2a 2= - ca +now add to both
b2a 2 sides.
The answer is D
Back to problemBack to problem
1) Do completi
ng the square
1) Now add to both
sides2) Now you
have your answer.
Standard 19 Standard 19
Rules and Strategies•First of you will need to divide b/a by 2 then you have to find the square root. Then add or subtract that number to both sides.• You’re completing the square.
Back to problemBack to problem
Standard 20 Students use the quadratic formula to find the roots of a second-
degree polynomial and to solve quadratic equations.
Problem 65
Problem 67
Problem 66
Problem 68 Main Menu
65) which is one of the solutions to the equation 2x2-x-4=0 ?
A. 1-√33 4
B. -1 +√33 4
C. 1+√33 4
D. -1-√33 4
Vocabulary Rules & Strategies
Solution & Answer Standard 20
Vocabulary
• Solutions: the answer itself• Equation: A mathematical statement that two
expressions are equal
Back to Problem
Rules & Strategies
• Identify a, b, and c • Use the quadratic formula -b+ √b2-4ac to solve
this equation 2(a)
Back to Problem
Solution
2x2-x-4 A=2 -b + √b2-4ac 1-4(-8)B=-1 2(a) 1+32C=-4 -b+ √(-1)-4(20(4) 2(2) 33 1+√33 4 Answer : C
1+√33 1-√33 4 4 Back to
Problem Standard 20
66.)Which statements best explains why there is no real solution to the quadratic
equation 2x2 + x + 7 = 0?
Vocabulary Rules and Strategies
Solution and Answer Standard 20
Vocabulary
• Statement ; declaration of speech setting forth facts, particulars, etc.
• Solution ; an explanation or answer• Quadratic Equation ; ax 2 + bx + c = 0
Back to Problem
Rules and Strategies
• Identify a, b , and c in ax 2 + bx + c• Find the discriminant : b 2 – 4ac• X>0 ; 2 solutions• X=0 ; 1 solution• X<0 ; no solution
Back to Problem
Solution and Answer 2x 2 + x + 7 = 0Identify:a= 2b= 1c= 7Discriminant:b 2 – 4ac(1)2 – 4 2 7∙ ∙ =1 – 56 = - 55Negative!
Answer:C Back to Problem Standard 20
-55 < 0 ;No Solution
67) What is the solution set of the quadratic equation 8x2+2x+1=0
A) -1 1 2, 4
B){-1+√2,-1 √2 }C) -1+√7, -1-√7
8 8D)No real solution
Vocabulary Rules & Strategies
Solution & Answer Standard 20
Vocabulary
• Solution: the process of determining the answer to a problem
• Quadratic Equation: an equation containing a single variable of degree 2. Its general form is ax 2 + bx + c = 0, where x is the variable and a, b, and c are constants ( a ≠ 0).
Back to Problem
Rules & Strategies
• Quadratic equation = ax2+bx+c=0• Plug in the values into the Quadratic equation• Simplify if possible• Answer
Back to Problem
Solution
X=-b±√b2-4ac 2(a)
x=-2±√4-32 16 x=-2±√-28 Negative
16 = ΦAnswer: D
Back to Problem Standard 20
68). What are the solutions to the equation 3x + 3 = 7x
A 7+ 85 or 7 - 85
B -7+ 85 or -7 - 85
C 7 + 13 or 7 - 13
D -7 + 13 or -7 - 13
Vocabulary Rules & Strategies
Solution & Answer Standard 20
2
6 6
6 6
6 6
6 6
Vocabulary
• Solution = answer
• Inequality = a mathematical statement that two expressions are equal.
Back to Problem
Rules & Strategies
Back to Problem
+ 2
Solution 3x + 3 = 7x Inverse operation to get in correct form
-7x3x – 7x + 3 = 0 a=3, b=-7, c=3
7 (-7) – 4(3)(3)2(3)
7 136
Answer: C Back to Problem Standard 20
=x
+
+
2
2
2
Standard 21 Students graph quadratic functions and know that their roots
are the x-intercepts.
Problem 69 Problem 71Problem 70
Problem 72 Main Menu
69) The graph of the equation y=x²-3x-4 is shown below:
For what values or value of x is y=0?A x=-1 onlyB x=-4 onlyC x=-1 and x=4D x=1 and x=-4
Vocabulary Rules & Strategies
Solution & Answer Standard 21
Vocabulary
• Equation= two equally balanced expressions
• Value= quantity
Back to Problem
Rules & Strategies
• Identify x-intercept(s), root(s), zero(s), solution(s)
Back to Problem
Solution
Solution:-1,4 Make sure to go
from left to right. Answer: C
Back to Problem Standard 21
x-intercept, root, solution, zeros x-intercept, root,
solution, zeros
70) Which best represents the graph of y= -x²+3?
a. b.
c. d.
VocabularyVocabulary Rules and StrategiesRules and Strategies
SolutionSolutionStandard
21Standard
21
Vocabulary
• Parabola = graph dervived from a quadratic function.
Back to ProblemBack to Problem
Rules and Strategies
• After each step, make sure you graph before moving onto the next one.
• Follow the steps: 1)Axis of Symmetry2)Vertex3)Y-Intercept4)Two Other Points
Back to ProblemBack to Problem
Solution and AnswerAxis of Symmetry = 0 Vertex = (0,3) Y intercept= y = -x²+3B=0 -(0) y= -(0)²+3 y= 3A= -1 2(-1) y=3
2 Other Points x= -1 x=-2Y= -(-1)²+3 y = -(-2)²+3Y= -1+3 y=-4+3y= 2 y= -1
Answer : B
Standard 21
Standard 21
Back to ProblemBack to Problem
71) Which quadratic function when graphed has x-intercepts of
4 and -3?
A y= (x-3)(x+4) B y= (x+3) (2x-8)
C y= (3x-1) (3x+1) D y= (3x+1) (8x-2)
Vocabulary Rules & Strategies
Solution & Answer Standard 21
Vocabulary
• X-intercepts: x-coordinates of the point where a graph intersects the x-axis.
• Quadratic function: A function that can be written in the form of f(x)= ax+bx=c where a, b, and c are real numbers and a zero.
Back to Problem
Rules & Strategies
• Do zero product property.• Check your work.
Back to Problem
Solutiony= (x-3)(x+4) y= (x+3)(2x-8) (x-3) (x+4)=0 (x+3)(2x-8)=0 x-3=0 x+4=0 x+3=0 2x-8=0 +3 +3 -4 -4 -3 -3 +8 +8 x=3 x=-4 x=-3 2x=8 3,-4 -3,4 2 2 x=-3 x=4 -3,4 = -3,4
Answer: BBack to Problem Standard 21
72) What are the real roots of the function in the graph?:
A 3B -6C -1 and 3D -6, -1, and 3
Vocabulary Rules & Strategies
Solution & Answer Standard 21
Vocabulary
• Roots: solution to function
Back to Problem
Rules & Strategies
• Look at the x-axis!• Identify x-intercept(s), root(s), zero(s),
solution(s)
Back to Problem
Solution
Solution:-1,3
Answer: c Back to Problem Standard 21
x-intercept, root, solution, zeros x-intercept, root,
solution, zeros
Standard 22 Students use the quadratic formula or factoring techniques or both to
determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points.
Problem 73 Main Menu
73) How many times does the graph of y = 2x2 – 2x + 3 intersect the x-axis?
a. Noneb. Onec. Twod. Three
Vocabulary Rules and Strategies
Solution and Answer
Standard 22
Vocabulary
• x-axis - the horizontal axis in a two-dimensional coordinate system
• Discriminant- the name given to the expression that appears under the square root (radical) sign in the quadratic formula.
Back to Problem
Rules and Strategies
• Try Super Diamond • Try discriminant
Back to Problem
Solution
Answer: ABack to Problem
y = 2x2 – 2x + 3Try :2x2 – 2x + 3 = 0
TRY DISCRIMINANT doesn’t worka=2 b2-4ac 2x +6 2x
b=-2 (-2)2-4(2)(3) -2
c=3 4-24 -20 No solution
Standard 22
Standard 23 Students apply quadratic equations to
physical problems, such as the motion of an object under the force of gravity.
Problem 74 Problem 75 Problem 76 Main Menu
74) An object that is projected straight downward with initial velocity v feet per second travels a distance s=vt+16t2 , where t= time in seconds.
If Ramon is standing on a balcony 84 feet above the ground and throws a penny straight down with initial velocity of 10 feet per second, in how many seconds will it reach the ground.
Vocabulary Rules & Strategies
Solution & Answer Standard 23
A) 2 secondsB) 3 secondsC) 6 secondsD) 8 seconds
Rules & Strategies
• Plug in 10 feet per second under v.• Plug in 84 as s.
Back to Problem
Vocabulary
• Velocity=the rate of speed with which something happens; rapidity of action or reaction.
Back to Problem
SolutionS=vt+16t 2
84=10t+16t 2 -84 -84
0=16t 2+10t-84 Gcf:2
2 2 2 2x2x2x2x3x7
8(-42) 0=2(8t 2 +5t-42) 8x 8x 0=2(t-2)(8t+21) -16 21 t-2=0 or 8t+21=0 5 t=2 or t= -21
8
You have to pick the one that’s most Logical. The answer is A)2. Back to
Problem Standard 23
1)Substitute known values into equation.2)Put it in standard form.3)Solve by doing gcf and then super diamond.4)Now do zero property.5) Find out which one makes more sense because time can`t be expressed as a negative it has to be 2 seconds.
1)Substitute known values into equation.2)Put it in standard form.3)Solve by doing gcf and then super diamond.4)Now do zero property.5) Find out which one makes more sense because time can`t be expressed as a negative it has to be 2 seconds.
75)The height of a triangle is 4 inches greater than twice its base. The area of the triangle is
168 square inches. What is the base of the triangle?
A) 7 in.B) 8 in.C) 12 in.D) 14 in.
Vocabulary Rules & Strategies
Solution & Answer Standard 23
Vocabulary• Height-The perpendicular distance from any vertex of a triangle to the side
opposite that vertex. Also called altitude. Sometimes the height is determined OUTSIDE of the triangle.
• Base:The side of a triangle to which an altitude is drawn. the base and the altitude will be used to find the area
• area-A = 1/2(bh), where b is the length of the base, and h is the length of the altitude. A = Square root [s(sa)(sb)(sc)], where s is the semiperimeter and a, b, and c are the lengths of the sides of the triangle.
• triangle-a three-sided polygon • Altitude-segment from the vertex of a triangle perpendicular to the line containing
the opposite side.• Vertex-the point of intersection of lines or the point opposite the base of a figure
Back to Problem
Rules & Strategies
• Find the height of the triangle• Plug in area for area formula• solve the formula• Use zero product property• FACTOR by GCF then Diamond !
Back to Problem
Solution
Back to Problem Standard 23
X+2 A=168
b
12 168
14
168=1\2b(2b+4)
2(168)=2b2+4b
0=2b2+4b-336
x
+
-168
2
b b
-12 +14
0=2(b-12)(b+14)
B-12=0B=+12
B+14=0 B=-14
answer: C
Step:1 Step: 2
Step: 3
Step: 4
Step: 5
Step: 6
76) A rectangle has a diagonal that measures 10 centimeters and a length that is 2 centimeters longer than the width. What is the width of the rectangle in centimeters?
a. 5b. 6c. 8d. 12
Vocabulary Rules and Strategies
Solution and Answer
Standard 23
Vocabulary
• Diagonal - a line joining two nonconsecutive vertices of a polygon or polyhedron
• Length – The measurement of the extent of something from the vertical side
• Width - The measurement of the extent of something from side to side
Back to Problem
Rules and Strategies
1. Draw a diagonal in the rectangle2. Use Pythagorean's theory to solve for the
width.A. a=2+w, w=widthB. b= wC. c=diagonal, 10cm
3. (2+w)2+w2= 102
Back to Problem
Visual Picture
w
2+w10
Solution (2+w)2+w2 = 102
(2+w)(2+w)+w2=100 4+4w+w2+w2=100 Combine like terms and put in descending -100 -100 order. 2w2+ 4w + -96 = Use super diamond method -192 w= 2 w 2 w =w 8 16 -12 -6 4 Answer: B
Back to Problem
(w+8)(w-6)= 0
Width = -8 or 6 width = 6
Standard 23
Standard 25.1 Students use properties of numbers to construct simple, valid
arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions.
Problem 23 Main Menu
23) John’s solution to an equation is shown below. Given: x+5x+6=0
Step 1: (x+2)(x+3)=0 Step 2: x+2=0 or x+3=0
Step 3: x= -2 or x= -3Which property of real numbers did john use for Step 2?
A multiplication property of equality. B zero product property of multiplication.
C commutative property of multiplication. D distributive property of multiplication over
addition.
Vocabulary Rules & Strategies
Solution & Answer Standard 25.1
Vocabulary
• Solution: the process of determining theanswer to a problem. • Equation: A mathematical statement that two
expressions are equal.• Zero Product Property: For real numbers p
and q, pq = 0 , then p= 0 or q =0 .
Back to Problem
Rules & Strategies
• Look at Step 2 and see which property it is.• Remember which property is which and don’t
mix them up.
Back to Problem
Solution
Step 2: x+2=0 or x+3=0Zero product property of
multiplication
Answer: BBack to Problem Standard 25.1
52) What is the perimeter of the figure shown below, which is not drawn to scale?
Vocabulary Rules & Strategies
Solution & Answer Standard 10
X+13
3x
8
X+5
2
3x+2
Vocabulary
• Perimeter: sum of all sides• Scale: size of the shape
Back to Problem
Rules and Strategies
• Add ALL of the sides • Combine like terms
Back to Problem
Solution and Answer
• 3x+2 + x+ 13+3x+8+2+x+5
• 8x + 30
• Answer: C
Back to Problem Standard 10
77) What is x2 – 4xy+ 4 y2 reduced to lowest terms? 3xy-6y2
A)x-2y C) x+2y 3 3B) x-2y D) x+2y 3y 3y
Vocabulary Rules & Strategies
Solution & Answer Standard 12
• Reduced: in simplest form
Vocabulary
Back to Problem
• Look at numerator and decide which factoring method is needed.
• Look at denominator and decide which factoring method is needed.
• Divide out common factors.
Rules and Strategies
Back to Problem
x2 – 4xy+ 4 y2 (diamond) +4 3xy-6y2 ( GCF) x x (x-2y)(x-2y) divide out!! -2 y -4 -2y 3y (x-2y) (x-2y) Answer: B 3xy -6y2
3y 3y 3y 3y (x-2y)
Solution and Answer
Back to Problem Standard 12