Difference Of Squares Perfect Square Trinomials - Spring-Ford Area .
FACTORING SHORTCUTS12.1.1Although most factoring problems can be done with generic rectangles, there are twospecial factoring patterns that, if recognized, can be done by sight. The two patterns areknown as Difference of Squares and Perfect Square Trinomials. The general patternsare as follows:Difference of Squares:Perfect Square Trinomial:a 2 x 2 b 2 ( ax b ) ( ax b )a 2 x 2 2abx b 2 ( ax b )2ExamplesDifference of SquaresPerfect Square Trinomialsx 2 49 (x 7)(x 7)x 2 10x 25 (x 5)24x 2 25 (2x 5)(2x 5)9x 2 12x 4 (3x 2)2x 2 36 (x 6)(x 6)x 2 6x 9 (x 3)29x 2 1 (3x 1)(3x 1)4x 2 20x 25 (2x 5)2Sometimes removing a common factor reveals one of the special patterns.Example 18x 2 50y 2 2(4x 2 25y 2 ) 2(2x 5y)(2x 5y)Example 212x 2 12x 3 3(4x 2 4x 1) 3(2x 1)2102 2006 CPM Educational Program. All rights reserved.Algebra Connections Parent Guide
ProblemsFactor each Difference of Squares.1.x 2 162.x 2 253.64m 2 254.4 p 2 9q 25.9x 2 y 2 496.x 4 257.64 y 28.144 25 p 29.9x 4 4y 2Factor each Perfect Square Trinomial.10.x 2 4x 411.y 2 8y 1612.m 2 10m 2513.x 2 4x 1614.a 2 8ab 16b 215.36x 2 12x 116.25x 2 30xy 9y 217.9x 2 y 2 6xy 118.49x 2 1 14xFactor completely.19.9x 2 1620.9x 2 24x 1621.9x 2 3622.2x 2 8xy 8y 223.x 2 y 10xy 25y24.8x 2 7225.4x 3 9x26.4x 2 8x 427.2x 2 8Answers1.( x 4 )( x 4 )2.( x 5 )( x 5 )3.4.( 2 p 3q ) ( 2 p 3q )5.( 3xy 7 ) ( 3xy 7 )6.7.( 8 y)( 8 y)8.(12 5 p ) (12 5 p )9.( 8m 5 ) ( 8m 5 )( x 5 )( x 5 )( 3x 2y )( 3x 2y )222210.( x 2 )211.( y 4 )212.( m 5 )213.not possible14.( a 4b )215.( 6x 1)216.( 5x 3y )217.( 3xy 1)218.( 7x 1)219.( 3x 4 ) ( 3x 4 )20.( 3x 4 )221.9 ( x 2 )( x 2 )22.2 ( x 2y )23.y ( x 5)224.8 ( x 3) ( x 3)25.x ( 2x 3) ( 2x 3)26.4 ( x 1)227.2 x2 42Chapter 12: Algebraic Extensions 2006 CPM Educational Program. All rights reserved.()103
ADDITION AND SUBTRACTION OFRATIONAL EXPRESSIONS12.1.2 and 12.1.3Addition and Subtraction of Rational Expressions uses the same process as simplenumerical fractions. First, find a common denominator (if necessary). Second, convertthe original fractions to equivalent ones with the common denominator. Third, add (orsubtract) the new numerators over the common denominator. Finally, factor thenumerator and denominator and reduce (if possible). See the Math Notes box on page503. Note that these steps are only valid provided that the denominator is not zero.Example 1The least common multiple of 2 ( n 2 ) and n ( n 2 )is 2n ( n 2 ) .To get a common denominator in the first fraction,multiply the fraction by nn , a form of the number 1.32(n 2) 3n(n 2) 3 n2(n 2) nMultiply the numerator and denominator of each term.It may be necessary to distribute the numerator. 3n2n(n 2) Add, factor, and simplify the result. (Note: n 0 or 2 ) 3n 62n(n 2) Multiply the second fraction by22 3 2n(n 2) 2.62n(n 2)3(n 2)2n(n 2) 32nExample 22(x 4)2 x 3x 62 x 3x 62x 8 2x 4 x 4x 4x 4x 4Example 333 x 23x 6 2x 2x 422 x 1 x 1 x 2x 1 x 2 x 2 x 1( x 1) ( x 2 )( x 1) ( x 2 )104 2006 CPM Educational Program. All rights reserved.Algebra Connections Parent Guide
ProblemsAdd or subtract each expression and simplify the result. In each case assume the denominatordoes not equal zero.1.3.5.7.9.11.13.15.17.19.x2 ( x 2 ) ( x 3) ( x 2 ) ( x 3)b2 9 22b 2b 3 b 2b 3x 10 x 6 x 2 x 23x 4 2x 5 3x 3 3x 36aa 1 225a a 5a a62x x(x 3) x(x 3)5x 6 5 x2x10a3 2a 6a 3a 185x 96 22x 2x 3 x 7x 123x 13x 5 22x 16 x 8x 162.4.6.8.10.12.14.16.1820.x4 2x 6x 8 x 6x 822a2 2a 2a 1 a 2a 1a 2b 2a b a ba b3x9 4x 12 4x 12x 2 3x 5 x 2 2x 10 101053 x 7 4(x 7)2x 4 2x 4 x 163x2 22x 8x x 4x 4x 5 22x 3x 28 x 2x 357x 16x 22x 2x 3 x x 22Answers1.1x 32.1x 23.5.26.37.9.13.17.19.x 32114.x 45(x 2)5x 10 2(x 4) ( x 1) x 3x 41a6x210.4(5x 6)(x 4) ( x 4 )2Chapter 12: Algebraic Extensions 2006 CPM Educational Program. All rights reserved.11.15.18.20.b 3b 1132x9a 62a 138.42323 12.4(x 7) 4x 2877 16.2(x 4) 2x 81414 2(x 7) ( x 7 ) x 494.x 2x 2 2(x 3) ( x 2 ) x 5x 6105
WORK AND MIXTURE PROBLEMS12.2.1 and 12.2.2Work problems are solved using the concept that if a job can be completed in r units of1time, then its rate (or fraction of the job completed) is .rMixture problems are solved using the concept that the product(value or percentage) x (quantity) must be consistent throughout the equation.Examples of Work ProblemsJohn can completely wash and dry the dishesin 20 minutes. His brother can do it in 30minutes. How long will it take them workingtogether?With two inflowing pipes open, a water tankcan be filled in 5 hours. If the larger pipe canfill the tank alone in 7 hours, how long wouldthe smaller pipe take to fill the tank?Solution: Let t the time to complete the1task so is their rate together. John’s rate ist11and his brother’s rate is. Since they2030are working together we add the two ratestogether to get the combined rate.Solution: Let t the time for the smaller11pipe so is its rate. The combined rate ist51and the larger pipe’s rate is . Since they are7working together we add the two ratestogether to get the combined rate.11 1 . Solve using20 30 tThe equation isfraction busters:60t( 201 301 ) 60t ( 1t )3t 2t 605t 60t 12 minutes106 2006 CPM Educational Program. All rights reserved.1 1 1 . Solve using7 t 5fraction busters:The equation is35t( 17 1t ) 35t ( 15 )5t 35 7t35 2tt 17.5 hoursAlgebra Connections Parent Guide
Examples of Mixture ProblemsAlicia has 10 liters of an 80% acid solution.How many liters of water should she add toform a 30% acid solution?A store has candy worth 0.90 a pound andcandy worth 1.20 a pound. If the ownerswant 60 pounds of candy worth 1.00 apound, how many pounds of each candyshould they use?Solution: Use (%) ( liters ) (%) ( liters )Let x liters of water added so x 10 is newliters.Solution: Use ( ) ( lbs.) ( ) ( lbs.)Let x pounds of 0.90 candyso 60 x pounds of 1.20 candy.The equation isThe equation is( 0.8 ) (10 ) ( 0.3) ( x 10 ) .Multiply by 10 to clear decimals and solve.( 8 ) (10 ) ( 3) ( x 10 )80 3x 3050 3xx 16 23 liters0.90(x) 1.20(60 x) 1.00(60) .Multiply by 100 to clear decimals and solve.90(x) 120(60 x) 100(60)90x 7200 120x 6000 30x 1200x 4040 lbs. of 0.90 candy and20 lbs. of 1.20 candyNote: The second example above could also have been solved using two equations, wherex 0.90 candy and y 1.20 candy:x y 60 and 0.90x 1.20y 60.00ProblemsSolve each problem.1. Susan can paint her living room in 2 hours. Her friend Jaime estimates it would take him3 hours to paint the same room. If they work together, how long will it take them to paintSusan’s living room?2. Professor Minh can complete a set of experiments in 4 hours. Her assistant can do it in6 hours. How long will it take them to complete the experiments working together?3. With one hose a swimming pool can be filled in 12 hours. Another hose can fill it in16 hours. How long will it take to fill the pool using both hoses?Chapter 12: Algebraic Extensions 2006 CPM Educational Program. All rights reserved.107
4. Together, two machines can harvest a tomato crop in 6 hours. The larger machine can do italone in 10 hours. How long does it take the smaller machine to harvest the crop workingalone?5. Steven can look up 20 words in a dictionary in an hour. His teammate Mary Lou can lookup 30 words per hour. Working together, how long will it take them to look up 100 words?6. A water tank is filled by one pump in 6 hours and is emptied by another pump in 12 hours.If both pumps are operating, how long will it take to fill the tank?7. Two crews can service the space shuttle in 12 days. The faster crew can service the shuttle in20 days alone. How long would the slower crew need to service the shuttle working alone?8. Janelle and her assistant Ryan can carpet a house in 8 hours. If Janelle could complete thejob alone in 12 hours, how long would it take Ryan to carpet the house working alone?9. Able can harvest a strawberry crop in 4 days. Barney can do it in 5 days. Charlie would take6 days. If they all work together, how long will it take them to complete the harvest?10. How much coffee costing 6 a pound should be mixed with 3 pounds of coffee costing 4 a pound to create a mixture costing 4.75 a pound?11. Sam’s favorite recipe for fruit punch requires 12% apple juice. How much pure apple juiceshould he add to 2 gallons of punch that already contains 8% apple juice to meet hisstandards?12. Jane has 60 liters of 70% acid solution. How many liters of water must be added to form asolution that is 40% acid?13. How many pound of nuts worth 1.05 a pound must be mixed with nuts worth 0.85 a poundto get a mixture of 200 pounds of nuts worth 0.90 a pound?14. A coffee shop mixes Kona coffee worth 8 per pound with Brazilian coffee worth 5 perpound. If 30 pounds of the mixture is to be sold for 7 per pound, how many pounds of eachcoffee should be used?15. How much tea costing 8 per pound should be mixed with 2 pounds of tea costing 5 per pound to get a mixture costing 6 per pound?16. How many liters of water must evaporate from 50 liters of an 8% salt solution to make a25% salt solution?17. How many gallons of pure lemon juice should be mixed with 4 gallons of 25% lemon juiceto achieve a mixture which contains 40% lemon juice?18. Brian has 20 ounces of a 15% alcohol solution. How many ounces of a 50% alcohol solutionmust he add to make a 25% alcohol solution?108 2006 CPM Educational Program. All rights reserved.Algebra Connections Parent Guide
Answers1.1.2 hours2.2.4 hours3.6 67 6.86 hours4.15 hours5.2 hours6.12 hours7.30 days8.24 hours9.6037 1.62 daysgallon12.45 liters14.20 Kona,10 Brazilian15.1 pound17.1 gallon18.8 ounces10.1.8 pounds11.13.50 pounds16.34 liters111Chapter 12: Algebraic Extensions 2006 CPM Educational Program. All rights reserved.109
FACTORING SHORTCUTS 12.1.1 Although most factoring problems can be done with generic rectangles, there are two special factoring patterns that, if recognized, can be done by sight. The two patterns are known as Difference of Squares and Perfect Square Trinomials. The general patterns are as follows: Difference of Squares: a 2x b2 ()ax b ()ax b
of perfect squares and square roots, concretely, pictorially and symbolically (limited to whole numbers). Squares & square roots Perfect squares Finding square roots 2. Determine the approximate square root of numbers that are not perfect squares (limited to whole numbers). Estimate square roo
Since 7 and 8 are consecutive numbers, there are no perfect squares between 49 and 64, which means that 50 is not a perfect square. 64 82 79: The perfect squares closest to 79 are 64 (64 82) and 81 (81 92). Since 8 and 9 are consecutive numbers, there are no perfect squares between 64 a
Michael Daniels Mostly Perfect Page 6 ‘Most-Perfect’ Magic Squares Mostly Perfect uses a 4x4 example from the subclass of most-perfect magic squares. In addition to summing to T across the rows, columns and diagonals, Zmost-perfect [ magic squares 5 have the additional mathematical properties that:6 The squares a
Section 6.3 Approximating Square Roots 247 EXAMPLE 2 Approximating Square Roots Estimate — 52 to the nearest integer. Use a number line and the square roots of the perfect squares nearest to the radicand. The nearest perfect square less than 52 is 49. The nearest perfect square greater than 52 is 64. Graph 52 . 49 7 64 8
When approximating square roots of numbers, you need to determine: · Between which two perfect squares it lies (and therefore which 2 square roots). · Which perfect square it is closer to (and therefore which square root). Example: Lies between 100 & 121, closer to 100. So is between 10 & 11, closer to 10. Square Perfect Root Square
Grade 9 Math Unit 1: Square Roots and Surface Area. Review from Grade 8: Perfect Squares What is a perfect square? Perfect square numbers are formed when we multiply a number (factor) by itself, or square a number. For Example: 9 is
Grade 8 Math Year End Review 2015 Unit 1: Square Root and Pythagorean Theorem 1. What is a perfect square?List the first 15 Perfect Squares. 2. Do you know the difference in squaring a number vs Square rooting a number?X2 vs ! Solve the following: 362 and 36 What does it mean to square a number? What does it mean to square root a number? 3.
mastery of it : writing . In a recent survey, academic staff at the University identified the interrelated skills of essay-writing and reasoning as the two most important skills for success in higher education; when asked which skills students most often lacked, essay-writing was again at the top of their list. Needless to say, writing ability is also highly prized by employers. The purpose of .
