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CST Prep Part II MAIN 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 48 Problem 49 Problem 50 Problem 51 Main Menu Problem 52 Rules & Strategies Vocabulary 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 B C D Solution & Answer 3x 2x² 3x² 2x³ 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 1 powers • Add exponents ( x = x ) x 2x Back to Problem Solution 2x • 1x 1 2x 1 2 Answer: B Back to Problem Standard 10 Vocabulary Solution and Answer 50) Which of the following expressions is equal to (x+2)+(x-2)(2x+1)? Rules and Strategies Standard 10 Solution and Answers (x+2)+(x-2x)(2x+1)? Step 1: Area Method 2x + 1 x -2 Step 2: Combine like terms and put in Descending order Answer: 2x² - 2x 2x² +x -4x -2 2x²-3x-2 1X + 2 + 2x² - 3x - 2 0+ 2x² -2x Back to Problem Standard 10 Vocabulary Area Method: (ax + b)(ax + b) ax + b ax b aax abx abx bb aax + (abx + abx) + bb Descending Order: ordering terms from greatest to least (ex. 1x+ 2x² + 3x+1x³ + 9 = 1x³ + 2x² + 3x + 1x + 9) Back to Problem Rules & Strategies 1st. Use the Area Method 2nd. Compare like terms 3rd. Descending Order Back to Problem Rules & Strategies Vocabulary 49)The sum of the two binomials is 5x2-6x, If one of the binomials is 3x2-2x, what is the other binomial A 2x2-4x B 2x2-8x C 8x2+4x D 8x2-8x Solution and Answer Standard 10 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 1st binomial 2nd binomial 2 (3x -2x) 2 +2x -4x 2 5x -6x Answer: A Sum of binomials Back to Problem Standard 10 Rules & Strategies Vocabulary 3 5x 47) 7 10x A) 2x4 B)1 2x4 C) 1 5x4 D) X4 5 Solution and Answer Standard 10.0 Vocabulary • Quotient of Powers: simplify coefficients combine 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 Rules and Strategies Vocabulary 48.) (4x²-2x+8) – (x²+3x-2) A) 3x²+x+6 B) 3x²+x+10 C) 3x²-5x+6 D) 3x²-5x+10 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 (4x²-2x+8) – 1 (x²+3x-2) (4x²-2x+8)-x²-3x+2 (4x²-1x²) + (-2x-3x) + (8+2) 3x²-5x+10 Answer:D Back to Problem Standard 10 Standard 11 Students solve multistep problems involving linear equations and inequalities in one variable. Problem 53 Problem 54 Problem 55 Problem 56 Main Menu Rules & Strategies Vocabulary 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) 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 gcf 3a²-24ab + 48b² 3 3 3 3(a²-8ab+16b²) +16 keep factoring!! a a -4b -4b 3 (a-4b)2 Standard 11 Back to Solution: C -8 Problem Rules & Strategies Vocabulary 54) Which is the factor of x² – 11x + 24 A B C D Solution & Answer x+3 x-3 x+4 x-4 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: B Back to Problem Standard 11 Rules & Strategies Vocabulary 55) Which of the following shows 2 9t + 12t + 4 factored completely? 2 A (3t + 2) B (3t + 4) (3t +1) C (9t + 4) (t + 1) 2 D 9t + 12t + 4 Solution & Answer 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 •Make sure to simplify the fractions 2 9t + 12t + 4 +36 3t 2 3 3 9t +6 9t +6 3 3 3t 2 +12 Answer: A (3t+2) 2 Back to Problem Standard 11 Rules & Strategies Vocabulary 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)² D. 8(2-z)² Solution & Answer Standard 11 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) Standard 11 Back to Problem Standard 12 • Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms. Problem 77 Problem 81 Problem 78 Problem 79 Problem 80 Main Menu Rules and Strategies Vocabulary 78.) 6x2 + 21x + 9 4x2 - 1 Solution and Answer A)3(x+1) 2x-1 B) 3(x+3) 2x-1 C) 3(2x+3) 4(x-1) D) 3(x+3) 2x+1 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 6x2 + 21x + 9 4x2 – 1 Solution and Answer 1) GCF 2) Difference of Two Squares 3)Cross Cancel Top/Numerator ; 6x2 + 21x + 9 Top/Numerator ; 6x2: 2 ∙ 3 ∙ x ∙ x 21x: 3 ∙ 7 ∙ x x = +2 9: 3 ∙ 3 x +3 GCF: 3 +6 +6 ∙ + +2 x+1 +7 Bottom/Denominator ; 4x2 - 1 2x 2x 1 1 (2x+1)(2x-1) Cross Cancel 3(x+3)(2x+1) = (2x+1)(2x-1) 6x 2 +21 + 9 3(x+3)(2x+1) 3 3 3 3 ( 2x 2 + 7x + 3) Super Diamond!! Back to Problem Answer ; B 3(x+3) 2x-1 Standard 12 Rules & Strategies Vocabulary 79)What is x2-4x+4 reduced to lowest terms? x2-3x+2 Solution & Answer A) x-2 X-1 B) x-2 X+1 C) x+2 X-1 D) x+2 X+1 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 Back to Problem • 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. Solution ax2+bx+c=0 Step 1: x2-4x+4 x2-3x+2 Step 2: x2-4x+4 (x-2) (x-2) Diamond Method 4 x * x -2 -2 x2-3x+2 + -4 Answer: A Divide out! (x-2) (x-2) = (x-2) (x-2) (x-1) (x-1) (x-2) (x-1) 2 x * -2 x -1 + -3 Back to Problem Standard 12 Rules and Strategies Vocabulary 80. What is 12a3 – 20a2 reduced to lowest terms? 16a2 + 8a Solution and Answer A) a 2 B) 3a – 5 2a + 1 C) -2a 4 + 2a D) a (3a – 5) 2 (2a + 1) 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 12a3 - 20a 16a2 + 8a Solution and Answer 2 2 4 a2 (3a – 5) 8 a (2a + 1) a (3a – 5) 2 (2a +1) Answer:D GCF:4a GCF: 8a2 Numerator : 12a3 - 20a2 4a2 4a2 4a2 (3a – 5) *Simplify Denominator: 16a2 + 8a 8a 8a 8a (2a + 1) Back to Problem Standard 12 Rules & Strategies Vocabulary 81) What is the simplest for of the fraction: _x2-1_ x2+x-2 Solution & Answer A)_-1__ x-2 B)x-1 x-2 C)x-1 x+2 D)x+1 x+2 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: 2 x -1 (x-1)(x+1) 2 x +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 1.GCF 2.Diamond 3.SuperDiamond 4.Difference of 2 Squares *make sure to separate work to make your work easier to read. Back to Problem Solution WORK: _x2-1_ x2+x-2 (x-1)(x+1) (x-1)(x+2) Answer:D Step 1) difference of 2 squares/diamond Difference of 2 squares: x2- 1 x x 1 1 (x-1)(x+1) Diamond: (x-1)(x+2) -2 x x -1 +2 +1 Back to Problem Standard 12 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 83 Problem 85 Main Menu Problem 84 Vocabulary Rules & Strategies 82) 7z²+7z • z²-4 4z+8 z³+2z²+z a. 7(z-2) 4(z+1) c. 7z(z+1) 4(z+2) Solution & Answer b.7(z+2) 4(z-1) d. 7z(z-1) 4(z+2) 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 GCF:7z -> GCF: 4 -> Solution & Answer 7z²+7z • z²-4 <- Difference of 2 Squares 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) Standard 13 Answer: A = 7(z-2) 4(z+1) Back to Problem Rules & Strategies Vocabulary 2 84) x+8x+16 x+3 2x+8 2 x-9 2 A 2(x+4) 2 (x-3)(x+3) B 2(x+3)(x-3) x+4 Solution & Answer part 1 C (x+4)(x-3) 2 2 D (x+4)(x-3) 2(x+3) Standard 13 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 Change to multiplication and take reciprocal of second rational expression LOOK BEFORE YOU BEGIN! 2 x+8x+16 x+3 x+8x+16 W +16 O R K 1x +4 . + 1x +4 2 2 2x+8 x+8x+16 x-9 Difference of 2 2 x-9 x+3 2x+8 Greatest Common 2 Factor: 2 2x+8 x-9 2x: 2 x xx33 (x+4)(x+4) (x+3)(x-3) 8: 2 2 2 (x+3) (x-3) x+3 2(x+4) 2 GCF: 2 2x+8 2 2 +8 (x+4) Solution Diivide out! (x+4) (x-3) 2 answer: C 2(x+4) Back to Problem Standard 13 Rules & Strategies Vocabulary 85) Which fraction is equivalent to 3x 5 x x 4+ 2 A. Solution & Answer X² 5 B. 9x² 20 c. 4 5 D. 9 5 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 & Answer Answer: C. 3x 5 x x 4+ 2 x x 4=4 x 2x +2=4 3x 4 3x 4 5 3x find Common denominator = 4 take reciprocal 4 5 Standard 13 Back to Problem Rules & Strategies Vocabulary 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 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 fraction Binomials: 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 and last terms : 2x² + 15 • Now, since numbers 2 and 3 are like terms combine them and put them 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 +2x 2x² 10x 3x² -15x ² -3x +2 +5 -5 -3 +3x +x +x 2x -10 -15 3x²-13x-10 2x²+7x-15 Combine 2x²+7x-15 on the top and 3x²-13-10on the bottom Answer: D Back to Problem Standard 13 Standard 14 Students solve a quadratic equation by factoring or completing the square. Problem 57 Problem 60 Problem 62 Problem 58 Problem 61 Problem 59 Main Menu Rules & Strategies Vocabulary 59) What are the solutions for the 2 quadratic equation x + 6x = 16? A -2,-8 B -2, 8 C 2,-8 D 2,8 Solution & Answer Standard 14 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 -16 x +8 +6 ax 2+ bx + c = 0 x 2 +6x = 16 x -16 -16 2 x2 + 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 Answer: C Back to Problem Standard 14 Rules & Strategies Vocabulary 2 57) If x is added to x, the sum is 42. Which of the following could be the value of x? A B C D Solution & Answer -7 -6 14 42 Standard 14 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 2 • Set up the expression (x) + x Back to Problem Solution Try x=-7 2 (x) + x= 42 2 (-7) +(-7)= 42 49 + (-7)=42 42=42 Answer: A Back to Problem Standard 14 Rules & Strategies Vocabulary 62) What are the solutions for 2 the quadratic equation x -8x=9? A) 3 B) 3,-3 C) 1,-9 D) -1,9 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 -9 -9 Step 2: 1x2-8x-9=0 Diamond Method -9 x * x +1 -9 Step 3: (x-9) (x+1) = 0 Zero Product X-9=0 x+1=0 Property +9=9 -1=-1 x=9 x=-1 Answers + -8 Answer: D Back to Problem Standard 14 Rules & Strategies Vocabulary 58) What quantity should be added to both sides of this equation to complete the square? x²-8x=5 A. 4 B. -4 C. 16 D. -16 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. 16 Standard 14 Back to Problem Rules & Strategies Vocabulary 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 Solution & Answer Standard 14 Vocabulary • Completing the square : a method for solving quadratic equations by using steps such as 1. Making sure you have the formula x² + bx + c 2. Find b ² 2 3. Completing the square with the answer to by adding it to both sides 4. 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 Completing the Square = b = 4 = 2 ² = 4 add to both sides of the equation 2 + 4x + 4 =6+4 factor by diamond method Perfect Square Trinomial : X² 2 factor the right, simplify the left Final Answer : ( x + 2 )² = 10 take square root of both sides Answer: B Back to Problem Standard 14 Vocabulary Rules & Strategies 61) Carter is solving this equation by factoring. 10x²-25x+15=0 Which expression could be one of his correct factors? a. x+3 b.x-3 c.2x+3 d. 2x-3 Solution & Standard 14 Answer 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 & Answer GCF: 5 10x²-25x+15=0 5 5 5 5 +6 5 ( 2x² -5x +3) =0 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. Back to Standard 14 Answer: D Problem Standard 15 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. Problem 85 Problem 86 Problem 88 Problem 89 Problem 90 Problem 91 Problem 87 Main Menu Rules & Strategies Vocabulary 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 hour B 60 miles per hour C 65 miles per hour D 70 miles per hour 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 Solution d = 45. 4 d = 180 180 = r . 3 isolate rate 3 3 60 = r Answer: B Back to Problem Standard 15 Rules & Strategies Vocabulary 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) B) C) D) Solution & Answer 60mL 45mL 40mL 25mL Standard 15 Vocabulary Distributive Property: is when you distribute what is outside of the parentheses. Isolate the variable: is to simplify the equation using operations to get the variable alone , on one side of the equal sign. Back to Problem Solution x +y 12% 100 mL 10% + 15% = 12% x + y = 100mL = .12(100)=12 =100mL .10x + x+y= 100 isolate y -> y=(100-x) 0.10x+0.15y=12 0.10x+ 0.15(100-x)=12 0.10x+15-0.15x=12 -15 .15y Work -0.15x +0.10x -0.05x=-3 -0.05 -0.05x -0.05 -15 0.10x-0.15x= -3 -0.05x=-3 isolate x X=60mL Answer: A Standard 15 Back to Problem 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 Problem Rules and Strategies Vocabulary 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 ? Solution and Answer a. 50min b. 25min c. 15min d. 12min Standard 15 Vocabulary • None available Back to Problem Rules & Strategies Find the number of minutes it takes to fill the tank together. There are 60 minutes in 1 hour Put information into fraction (over 1) and find the LCM. Once LCM is found, distribute the LCM inside whatever is in the parentheses . Back to Isolate “m” and your answer Problem will be found. Solution 1 1 20m + 30m = 1 LCM: LCM: m=minutes 20: 22 5 Choose one with the greatest power 30: 2 5 3 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 Rules & Strategies Vocabulary 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 2.5 B 4 C 5 D 10.8 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 Rules & Strategies Vocabulary 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. B. C. D. Solution 0.4 liter 0.5 liter 2 liters 8 liters 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 Answer x= 25 % = .25 y= 100% because its it's PURE fruit juice = 1 Solve for y Results in 2 liters of 10% = (.10) fruit juice First, isolate x then plug it in. x-y = 2 .25x - 1y = 2(.10) x-y=2 (2.4) – y = 2 -2.4 -2.4 -y = -.4 -1 -1 -y=2-x -1 -1 -1 y= -2+x .25x-(-2+x)=.2 .25x+2-x=.2 -2 -2 .25x-x= -1.8 -.75x = -1.8 -.75= -.75 y= .4 x = 2.4 Standard 15 Back to Problem Rules & Strategies Vocabulary Miles traveled 600 450 300 960 Gallons of gasoline 20 15 10 x 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 B C D Solution & Answer 32 41 55 80 Standard 15 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 16 16. 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 Rules& Strategies Vocabulary 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 B C D Solution & Answer 3 5 6 8 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 Add all of them to make the expression equivalent to 38 Back to Problem Solution x+(x-2) + 2x + 2(2x) = 38 x + x - 2 + 2x + 4x = 38 +2 +2 8x = 40 8x = 40 8 8 Beth is five years x=5 Combine like terms old. Answer: B Back to Problem Standard 16 Rules & Strategies Vocabulary 93) Which relation is a function? A.Input Output B.Input Output C.Input Output D.Input Output 1 2 2 6 1 2 0 1 2 2 2 5 2 4 0 2 3 3 6 4 4 6 1 3 4 3 6 3 4 8 1 4 Solution & Answer Standard 16 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 Input Output 1 2 2 2 3 3 4 3 Answer: A Back to Problem Standard 16 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 Rules and Strategies Vocabulary 94) For which equation graphed below are all the y-values negative? a. Solution and Answer b. c. d. Standard 17 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 cross in the y-int of -1. Answer: A 4 1 3 2 Back to Problem Standard 17 Vocabulary Rules & Strategies 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} Solution & Answer Standard 17 Vocabulary • Domain=the set of all first coordinates (xvalues) 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 *Only focus on domain Solution Answer: D; {1,2,4,5} X y 1 -1 2 -2 4 -4 5 -5 Back to Problem Standard 17 Standard 18 18. 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 96 Main Menu 96) Which of the following graphs represents a relation that is not a function of x? Vocabulary Solution and answer Click here to look at graphs. Rules and Srategies Standard 18 A Back to problem B C D vocabulary 1. Function_ : A relation ship or expression involving one or more variables. (y=mx+b) 2. Vertical line test: putting strait lines going through X Axis. It tests to see Whether it is a function or not. Back to problem Solution and answer Remember To Give each a vertical line test D, isn`t a function Because it is being hit more than once. Back to problem Standard 18 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 problem Standard 19 Students know the quadratic formula and are familiar with its proof by completing the square. Problem 63 Main Menu Problem 64 Rules & Strategies Vocabulary 64) Four steps to derive the quadratic formula are shown below. What is the correct order for these steps? Solution & Answer A) B) C) D) I, IV, II, III I, III, IV, II II, IV, I, III II, III, I, IV 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 Vocabulary #63 Rules and Srategies Toni is solving this equation by completing the square. Step1: ax2 + bx =-c Step 2: x2 + ba X= - ca Step 3:? Which should be step 3 in the solution? A)X2 =-cb - baX B)X +ba = - cax C)X 2 + baX+b2a = -ca + b2a D)X 2 +baX+ b2a 2 = - ca + b2a 2 Solution and answer Standard 19 Vocabulary 1. 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 problem Solution and Answer Step1: ax2 + bx =-c b/a 2 Step 2: x2 + ba X= - ca 2 b x 1 2= b Step 3:? a 2 2a 2 x2 + ba X+ b2a 2= - ca +now add to both b sides. 2a 2 1) Do completi ng the square 1) Now add to both sides 2) Now you have your answer. The answer is D Standard 19 Back to problem 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 problem Standard 20 Students use the quadratic formula to find the roots of a seconddegree polynomial and to solve quadratic equations. Problem 65 Problem 66 Problem 67 Problem 68 Main Menu Rules & Strategies Vocabulary 65) which is one of the solutions to the equation 2x2-x-4=0 ? A. B. C. Solution & Answer D. 1-√33 4 -1 +√33 4 1+√33 4 -1-√33 4 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=-1 C=-4 -b + √b2-4ac 1-4(-8) 2(a) -b+ √(-1)-4(20(4) 1+32 2(2) 1+√33 33 4 Answer : C 1+√33 1-√33 4 4 Back to Problem Standard 20 Vocabulary Rules and Strategies 66.)Which statements best explains why there is no real solution to the quadratic equation 2x2 + x + 7 = 0? A)The value of 12 – 4 ∙ 2 ∙ 7 is positive. B)The value of 12 – 4 ∙ 2 ∙ 7 is equal to 0. C)The value of 12 – 4 ∙ 2 ∙ 7 is negative. D)The value of 12 – 4 ∙ 2 ∙ 7 is not a perfect square. 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 = 0 Identify: a= 2 b= 1 c= 7 Discriminant: b 2 – 4ac (1)2 – 4 ∙ 2 ∙ 7 =1 – 56 = - 55 Negative! Answer:C -55 < 0 ; No Solution Back to Problem Standard 20 Rules & Strategies Vocabulary 67) What is the solution set of the quadratic equation 8x2+2x+1=0 A) -1 2, Solution & Answer 1 4 B){-1+√2,-1 √2 } C) -1+√7, -1-√7 8 8 D)No real solution 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 2 ax + 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 =Φ Answer: D Back to Problem Standard 20 Rules & Strategies Vocabulary 68). What are the solutions to the equation 2 3x + 3 = 7x A 7+ 85 or 7 - 85 6 6 B -7+ 85 or -7 - 85 6 6 C 7 + 13 or 7 - 13 6 6 D -7 + 13 or -7 - 13 6 Solution & Answer 6 Standard 20 Vocabulary • Solution = answer • Inequality = a mathematical statement that two expressions are equal. Back to Problem Rules & Strategies • • • • • Plug in correctly Take both sides of the square Solve for “x” Quadratic-identify a, b, and c 2 + Quadratic = -b b – 4ac 2a Back to Problem Solution 3x2+ 3 = 7x Inverse operation to get in correct form -7x 3x – 7x 2+ 3 = 0 a=3, b=-7, c=3 + (-7)2 – 4(3)(3) 7 x= 2(3) 7 + 13 6 Answer: C Back to Problem Standard 20 Standard 21 Students graph quadratic functions and know that their roots are the x-intercepts. Problem 69 Problem 70 Problem 72 Main Menu Problem 71 Rules & Strategies Vocabulary 69) The graph of the equation y=x²-3x-4 is shown below: For what values or value of x is y=0? Solution & Answer A B C D x=-1 only x=-4 only x=-1 and x=4 x=1 and x=-4 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 Rules and Strategies Vocabulary 70) Which best represents the graph of y= -x²+3? a. c. Solution b. d. Standard 21 Vocabulary • Parabola = graph dervived from a quadratic function. Back to Problem Rules and Strategies • After each step, make sure you graph before moving onto the next one. • Follow the steps: 1) Axis of Symmetry 2) Vertex 3) Y-Intercept 4) Two Other Points Back to Problem Solution and Answer Axis of Symmetry = 0 B=0 -(0) A= -1 2(-1) Vertex = (0,3) y= -(0)²+3 y=3 Y intercept= y = -x²+3 y= 3 2 Other Points x= -1 x=-2 Y= -(-1)²+3 y = -(-2)²+3 Y= -1+3 y=-4+3 y= 2 y= -1 Back to Problem Answer : B Standard 21 Rules & Strategies Vocabulary 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) 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 Solution y= (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: B Back to Problem Standard 21 Rules & Strategies Vocabulary 72) What are the real roots of the function in the graph?: Solution & Answer A B C D 3 -6 -1 and 3 -6, -1, and 3 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 Standard 22 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the xaxis in zero, one, or two points. Problem 73 Main Menu Rules and Strategies Vocabulary 73) How many times does the graph of y = 2x2 – 2x + 3 intersect the x-axis? a. b. c. d. Solution and Answer None One Two Three Standard 22 Vocabulary • x-axis - the horizontal axis in a twodimensional 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 y = 2x2 – 2x + 3 Try : 2x2 – 2x + 3 = 0 TRY DISCRIMINANT doesn’t work a=2 b2-4ac b=-2 (-2)2-4(2)(3) c=3 2x +6 2x -2 4-24 -20 No solution Answer: A Back to Problem 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 Rules & Strategies Vocabulary 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. Solution & Answer A) 2 seconds B) 3 seconds C) 6 seconds D) 8 seconds Standard 23 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 Solution S=vt+16t 2 84=10t+16t 2 -84 0= -84 16t 2+10t-84 2 2 2 Gcf:2 2x2x2x2x3x7 8(-42) 0=2(8t 2 +5t-42) 8x 0=2(t-2)(8t+21) -16 8x 21 t-2=0 or 8t+21=0 5 t=2 or t= -218 You have to pick the one that’s most Logical. The answer is A)2. 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. Back to Problem Standard 23 Rules & Strategies Vocabulary 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. 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 -168 b -12 Step: 6 x b + +14 Solution 14 X+2 2 A= 168 12 168 Step: 2 168=1\2b(2b+4) Step: 3 0=2(b-12)(b+14) B-12=0 B=+12 B+14=0 B=-14 b 2(168)=2b2+4b Step: 4 0=2b2+4b-336 Step: 5 answer: C Back to Problem Standard 23 Rules and Strategies Vocabulary 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. b. c. d. Solution and Answer 5 6 8 12 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 rectangle 2. Use Pythagorean's theory to solve for the Visual Picture width. A. a=2+w, w=width B. b= w C. c=diagonal, 10cm 10 2+w 3. (2+w)2+w2= 102 w Back to Problem 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 (w+8)(w-6)= 0 -192 w= 2 w 2 w =w 8 16 -12 -6 Width = -8 or 6 width = 6 4 Answer: B Back to Problem 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 Rules & Strategies Vocabulary 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= -3 Which property of real numbers did john use for Step 2? A B C D Solution & Answer multiplication property of equality. zero product property of multiplication. commutative property of multiplication. distributive property of multiplication over addition. Standard 25.1 Vocabulary • Solution: the process of determining the answer 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=0 Zero product property of multiplication Answer: B Back to Problem Standard 25.1 Rules & Strategies Vocabulary 52) What is the perimeter of the figure shown below, which is not drawn to scale? X+13 3x 3x+2 8 2 X+5 Solution & Answer Standard 10 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 Rules & Strategies Vocabulary 77) What is x2 – 4xy+ 4 y2 reduced to lowest terms? 3xy-6y2 A) x-2y 3 B) x-2y 3y Solution & Answer C) x+2y 3 D) x+2y 3y Standard 12 Vocabulary • Reduced: in simplest form Back to Problem Rules and Strategies • Look at numerator and decide which factoring method is needed. • Look at denominator and decide which factoring method is needed. • Divide out common factors. Back to Problem Solution and Answer x2 – 4xy+ 4 y2 (diamond) 3xy-6y2 ( GCF) (x-2y)(x-2y) divide out!! 3y (x-2y) (x-2y) Answer: B 3y +4 x -2 y -4 x -2y 3xy -6y2 3y 3y 3y (x-2y) Back to Problem Standard 12