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1.2 Radical Expressions and Equations𝒏Principle 𝒏-th root 𝒂:Let 𝑛 be a positive integer greater than 1, and let 𝑎 be a real number.𝑛(1) If 𝑎 0 then 𝑎 0𝑛(2) If 𝑎 0 then 𝑎 is the positive real number 𝑏 such that 𝑏 𝑛 𝑎𝑛(3) (a) If 𝑎 0 and 𝑛 is odd, then 𝑎 is the negative real number 𝑏 such that 𝑏 𝑛 𝑎𝑛(b) If 𝑎 0 and 𝑛 is even, then 𝑎 is not a real number.𝑛The expression 𝑎 is a radical, the number 𝑎 is the radicand, and 𝑛 is the index of the radical.The symbol is called a radical sign.3If 𝑎 𝑏, then 𝑏 2 𝑎. If 𝑎 𝑏, then 𝑏 3 𝑎𝒏Properties of 𝒂: (where 𝑛 is a positive integer)Example233( 7) 7, ( 5) 5Property𝑛( 𝑎 ) 𝑎 if 𝑎 is a real number𝑛𝑛𝑛3 𝑎𝑛 𝑎 if 𝑎 0 32 3, 23 23𝑛5 𝑎𝑛 𝑎 if 𝑎 0 and 𝑛 is odd ( 2)3 2, ( 3)5 34 ( 2)2 2 2, ( 3)4 3 3𝑛 𝑎𝑛 𝑎 if 𝑎 0 and 𝑛 is even 𝑥 2 𝑥 { 𝑥 if 𝑥 0 𝑥 if 𝑥 016

Laws of Radicals:Example 98 49 2 49 2 7 233333 54 ( 27)(2) 27 2 3 23 3 64 (2)(3)𝑛𝑎 𝑎 𝑛𝑏 𝑏3𝑛3 3 3 3 8 823Law𝑛 𝑎𝑏 𝑛 𝑎 𝑏𝑛66 64 64 26 2𝑚 𝑛 𝑎 𝑚𝑛 𝑎Warning: If 𝑎 0 and 𝑏 0Warning 𝑎2 𝑏2 𝑎 𝑏 𝑎 𝑏 𝑎 𝑏Example 32 42 25 5 3 4 7 4 9 13 4 9 2 3 5Simplifying RadicalsAn expression involving radicals is in simplest form when the following conditions are satisfied:1. All possible factors have been removed from the radical2. All fractions have radical-free denominators (accomplished by a process calledrationalizing the denominator)3. The index of the radical is reduced.To simplify a radical, we factor the radicand into factors whose exponents are multiples of theindex. The roots of these factors are written outside the radical and the “leftover” factors makeup the new radicand.17

Example 1: Simplify3 135Factor 135 27 5Split into two radicals333 33 533 5Simplify the first radicalFinal answerExample 2: Simplify3 16𝑎3 𝑏 8 𝑐 4Factor into multiples of the index (23 𝑎3 𝑏 6 𝑐 3 )(2𝑏 2 𝑐)Divide a 3 (the index) out of each exponent33 (2𝑎𝑏 2 𝑐)3 (2𝑏 2 𝑐)33Split into two radicals (2𝑎𝑏 2 𝑐)3 2𝑏 2 𝑐Simplify first radical3Final answer2𝑎𝑏 2 𝑐 2𝑏 2 𝑐Example 3: Simplify18 3𝑥 2 𝑦 3 6𝑥 5 𝑦Multiply radicals together, factor the 6 3𝑥 2 𝑦 3 2 3𝑥 5 𝑦Combine bases 32 2𝑥 7 𝑦 4Factor into multiples of the index (32 𝑥 6 𝑦 4 )(2𝑥)Divide a 2 (the index) out of each exponent (3𝑥 3 𝑦 2 )2 (2𝑥)Split into two radicals (3𝑥 3 𝑦 2 )2 2𝑥Simplify the first radical3𝑥 3 𝑦 2 2𝑥Final answer

Rationalizing Denominators of Quotients (𝑎 0)Factor indenominator 𝑥3 𝑥7 𝑥 3Multiply numeratorand denominator by 𝑥Resulting factor 𝑥 𝑥 𝑥 2 𝑥33 𝑥 23 𝑥 𝑥 2 𝑥777 𝑥 3 𝑥 4 𝑥 𝑥 4Example 4: Simplify1Multiply numerator and denominator by 3 31 3( ) 3 3Multiply 3Simplify denominator 32 33Final answerExample 5: Simplify133Multiply numerator and denominator by 𝑎2 𝑎3Multiply 𝑎2Simplify denominator 𝑎2()33 𝑎 𝑎2133 𝑎33 𝑎2𝑎Final answer19

Example 6: Simplify3 2Square root of numerator and denominator 3Multiply numerator and denominator by 2 2 3 2( ) 2 2Multiply 6Simplify denominator 22 62Final answerExample 7: Simplify5 𝑥𝑦25 𝑥Take root of numerator and denominator5Multiply numerator and denominator by 𝑦 35 𝑦 255 𝑦 3()55 𝑦 2 𝑦 3 𝑥5 𝑥𝑦 3MultiplySimplify denominator5 𝑦 55 𝑥𝑦 3𝑦20Final answer

Definition of Rational Exponents:Let𝑚𝑛be a rational number where 𝑛 is a positive integer greater than 1. If 𝑎 is a real number such𝑛that 𝑎 exists, then𝑛(1) 𝑎1/𝑛 𝑎𝑛𝑛(2) 𝑎𝑚/𝑛 𝑎𝑚 ( 𝑎)𝑚𝑚(3) 𝑎𝑚/𝑛 (𝑎1/𝑛 ) (𝑎𝑚 )1/𝑛Example 8: Simplify( 125)2/3 (4) 5/223( ( 5)3 ) ( 4)Convert to radicals 5Take roots( 5)2 (2) 5Move negative exponent and make it positive( 5)225Evaluate exponents2532Final answerExample 9: Simplify(𝑎2 𝑏 9 )1/3Put exponent on each factor(𝑎2 )1/3 (𝑏 9 )1/3Multiply exponents𝑎2/3 𝑏 3Final answer21

Example 10: Simplify22𝑥 2/33𝑥 1/6( 1/2 ) ( 1/3 )𝑦𝑦Put exponent on each factor, squaring 2 to get 44𝑥 4/3 3𝑥 1/6() ( 1/3 )𝑦𝑦Multiply bases, add exponents(4 3)𝑥 4/3 1/6𝑦1 1/3Common denominators on exponents12𝑥 8/6 1/6𝑦 3/3 1/3Add exponents12𝑥 7/6𝑦 4/3Final answerAddition and Subtraction with RadicalsWe add or subtract radicals just as we add or subtract like termsExample 11: Add7 5 3 5Add like radicals, radical remains unchanged10 5Final answerExample 12: Add3339 2 7𝑥 2 4 23313 2 7𝑥 23Add like radicals, note 7𝑥 2 is not like othersFinal answerExample 13: Add5538 4𝑥 2 4𝑥 4𝑥5310 4𝑥 4𝑥22Add like radicals, note index must matchFinal answer

Example 14: Add35 16𝑦 4 7𝑦 3 2𝑦3Factor first radical5 23 𝑦 3 3 2𝑦 7𝑦 3 2𝑦Simplify first radical5 2𝑦 3 2𝑦 7𝑦 3 2𝑦Multiply10𝑦 3 2𝑦 7𝑦 3 2𝑦Add like terms17𝑦 3 2𝑦Final answerRadical EquationsThe principle of powers: If an equation 𝑎 𝑏 is true, then 𝑎𝑛 𝑏 𝑛 is true for any rationalnumber 𝑛 for which 𝑎𝑛 and 𝑏 𝑛 exist. This means we will have to check our solutions in theoriginal equation to be sure they work. If a value does not work it is called an extraneoussolution and not included in the final answerExample 15: Solve 5𝑥 3 7Square both sides5𝑥 3 49Subtract 3 from both sides5𝑥 46Divide both sides by 5𝑥 465Check solution46 5 ( ) 3 75 46 3 7 49 77 7𝑥 465It works!Final answer23

Example 16: Solve 4𝑥 9 2𝑥 0Add 2𝑥 to isolate one radical term 4𝑥 9 2𝑥Square both sides4𝑥 9 2𝑥Subtract 2𝑥 and add 9 to both sides2𝑥 9Divide by 2𝑥 92Check solution99 4 ( ) 9 2 ( ) 022 18 9 9 0 9 3 03 3 00 0𝑥 2492It works!Final answer

Example 17: Solve 3 𝑥 𝑥 3Add 𝑥 to isolate the radical 3 𝑥 𝑥 3Square both sides3 𝑥 𝑥 2 6𝑥 9Subtract 3 and add 𝑥0 𝑥 2 7𝑥 6Factor0 (𝑥 6)(𝑥 1)Set each factor equal to zero𝑥 6 0or𝑥 1 0Solve both equations𝑥 6, 1Check both solutions 3 ( 6) ( 6) 3 9 6 33 6 39 3Checking 𝑥 6Extraneous solution, not included in final answer 3 ( 1) ( 1) 3 4 1 32 1 33 3Checking 𝑥 1𝑥 1Final answerIt works!25

Example 18: Solve 4𝑦 1 𝑦 2 3Adding 𝑦 2 isolates one of the radical terms 4𝑦 1 3 𝑦 2Square both sides2( 4𝑦 1) (3 𝑦 2)Simplify, recall (𝑎 𝑏)2 𝑎2 2𝑎𝑏 𝑏 24𝑦 1 9 6 𝑦 2 𝑦 2Combine like terms4𝑦 1 7 𝑦 6 𝑦 2Subtract 7 and y to isolate the term with radical3𝑦 6 6 𝑦 2Divide by common factor of 3𝑦 2 2 𝑦 2Square both sides(𝑦 2)2 (2 𝑦 2)2622Simplify, recall (𝑎 𝑏)2 𝑎2 2𝑎𝑏 𝑏 2𝑦 2 4𝑦 4 4(𝑦 2)Distribute𝑦 2 4𝑦 4 4𝑦 8Subtract 4y, add 8 to make equal to zero𝑦 2 8𝑦 12 0Factor(𝑦 6)(𝑦 2) 0Set each factor equal to zero𝑦 6 0 or 𝑦 2 0Solve𝑦 6,2We need to check these answers 4(6) 1 (6) 2 3 24 1 6 2 3 25 4 35 2 33 3Checking 𝑦 6 4(2) 1 (2) 2 3 8 1 2 2 3 9 0 33 0 33 3𝑦 6,2Checking 𝑦 2It works!It works!Final answer

1.2 Radical Expressions and Equations PracticeSimplify each expression.1. 50 2 82. 63 2 7 273.2 2 54 2504.𝑥 2𝑥 4 18𝑥 35.1 826.1 48 2 37. 75 4 18 2 12 2 88.3 2𝑎3 𝑎 18𝑎 2 8𝑎39. 125 2 27 20 3 1210.12 12𝑥 3 𝑥311. 3𝑎 5 27𝑎3 𝑎 3𝑎12.1 3 27 2 12313.3 5 60 3 15514.3 50 4 8 27 315. 𝑥 4 𝑦 𝑥 9𝑥 2 𝑦 𝑥 2 16𝑦16.𝑎2 8𝑎3 𝑏 2𝑎3 18𝑎𝑏 3𝑎 50𝑎5 𝑏17. 16𝑎𝑏 3 9𝑎𝑏 3 25𝑎3 𝑏 318.95 25𝑥 𝑥𝑥 𝑥19.12 32 2920.712 189321.1 3 1 1 7 752 4 2 322.15 2 4 25 125 17 ( ) 5416333127

23.25.27.29.24.326 24 3 234 36 8(4) 3/2322 27 ( ) 2𝑥1 𝑥 𝑥 𝑥452 60326.4 9 28.4134 144 3 9 5 48 430.5 3 1 410032.33.231 241/2 481/332 634.𝑏𝑎3 𝑏𝑎2 𝑎435.𝑎 𝑏𝑎 𝑏 𝑎 𝑏𝑎 𝑏36.𝑎 111 1 2 1 2 (𝑎 ) ( )𝑎 1𝑎𝑎 𝑎3 431.337. 1 38.661 18 121/23 31 33 2 54 250 24 93𝑥𝑦 2 𝑥𝑦 2 𝑦𝑦2 3𝑥 2 𝑦 3 12𝑥 3 𝑦 27𝑥 5 𝑦 75𝑦39.1 1 18 108 121/2 33/2 5 33 6 340.1 12 2 25 24 6 96 66 6 23 5 641.51171 3(8)2/3 43 27 ( 27)2/3 1474 3263286

Solve each equation.42. 7𝑥 2 443. 2𝑥 3 3 044. 3𝑥 12 6 045. 5𝑥 1 4 046. 3𝑥 8 𝑥 047. 6𝑥 5 𝑥 048. 5𝑥 1 1 𝑥49. 𝑥 2 𝑥 250.𝑥 4𝑥 1 551.3 𝑥 6𝑥 1352. 5𝑥 1 4𝑥 4 053.𝑥 1 7 𝑥54. 6𝑥 8 3𝑥 4 055. 3 3𝑥 1 2𝑥56. 2𝑥 1 𝑥 157. 2𝑥 2 3 2𝑥 158. 3𝑥 2 𝑥 259. 4𝑥 5 𝑥 4 260. 3𝑥 1 4 𝑥 361. 3𝑥 4 𝑥 2 262. 𝑥 2 5 3𝑥 363. 2𝑥 4 𝑥 3 164. 4𝑥 4 𝑥 2 2𝑥 365. 3𝑥 4 2𝑥 1 𝑥 366. 4𝑥 1 𝑥 1 7𝑥 267. 2𝑥 1 𝑥 1 𝑥68. 6𝑥 2 2𝑥 6 15𝑥 1729

1.3 Quadratic Expressions and EquationsAlgebraic ExpressionsIt is convenient to use letters such as 𝑥 or 𝑦 to represent numbers. Such a symbol is called avariable. An algebraic expression is the result of performing a finite number of additions,subtractions, multiplications, divisions, or roots on a collection of variables and real numbers.The following are examples of algebraic expressions:𝑥 3 3𝑥 2 𝑥 𝜋,4𝑥𝑦 𝑥,𝑥 𝑦3 7𝑥 3 𝑧𝑥 5 𝑦 2PolynomialsIf 𝑥 is a variable, then a monomial in 𝑥 is an expression of the form 𝑎𝑥 𝑛 , where 𝑎 is a realnumber and 𝑛 is a nonnegative integer. A binomial is a sum of two monomials, and a trinomial isa sum of three monomials.Definition of a polynomial:A polynomial in 𝑥 is the sum of the form:𝑎𝑛 𝑥 𝑛 𝑎𝑛 1 𝑥 𝑛 1 𝑎1 𝑥 𝑎0Where 𝑛 is a nonnegative integer and each coefficient 𝑎 is a real number. If 𝑎𝑛 0, then thepolynomial is said to have degree 𝑛. Each expression 𝑎𝑘 𝑥 𝑘 in the sum is a term of thepolynomial. The coefficient 𝑎𝑘 of the highest power of 𝑥 is called the leading coefficient of thepolynomial.Example2𝑥 3𝑥 4 ( 7)𝑥 3Leading Coefficient2Degree5𝑥 7 8𝑥 2 ( 3)𝑥17 5𝑥 2 1 52880530

Example 1: Add(𝑥 3 4𝑥 2 5𝑥 9) (2𝑥 3 𝑥 2 3𝑥)Distribute “ ”𝑥 3 4𝑥 2 5𝑥 9 2𝑥 3 𝑥 2 3𝑥Combine like terms3𝑥 3 3𝑥 2 2𝑥 9Final answer(𝑥 3 5𝑥 2 10𝑥 6) (2𝑥 3 3𝑥 4)Distribute “ “𝑥 3 5𝑥 2 10𝑥 6 2𝑥 3 3𝑥 4Combine like terms 𝑥 3 5𝑥 2 7𝑥 10Final answerExample 2: SubtractExample 3: Multiply(7𝑥 5𝑦)(3𝑥 2𝑦)Use FOIL to multiply(7𝑥)(3𝑥) 21𝑥 2F – First terms(7𝑥)( 2𝑦) 14𝑥𝑦O – Outside terms(5𝑦)(3𝑥) 15𝑥𝑦I – Inside terms(5𝑦)( 2𝑦) 10𝑦 2L – Last terms21𝑥 2 14𝑥𝑦 15𝑥𝑦 10𝑦 2Combine like terms21𝑥 2 𝑥𝑦 10𝑦 2Final answerExample 4: Multiply(3𝑥 7)(2𝑥 3 3𝑥 1)(3𝑥)(2𝑥 3 ) (3𝑥)(3𝑥) (3𝑥)( 1) ( 7)(2𝑥 3 ) ( 7)(3𝑥) ( 7)( 1)6𝑥 4 9𝑥 2 3𝑥 14𝑥 3 21𝑥 76𝑥 4 14𝑥 3 9𝑥 2 24𝑥 7Distribute3𝑥 and 7MultiplyCombinelike termsFinal answerSpecial ProductsCertain products of binomials occur so frequently that you should learn to recognize them31

Special Product/Factor Formulas:NameDifference of two squaresProperty(𝑎 𝑏)(𝑎 𝑏) 𝑎2 𝑏 2𝑎2 𝑏 2 𝑃𝑟𝑖𝑚𝑒Sum of squares(𝑎 𝑏)2 𝑎2 2𝑎𝑏 𝑏 2(𝑎 𝑏)2 𝑎2 2𝑎𝑏 𝑏 2Perfect square trinomialSum of cubes(𝑎 𝑏)(𝑎2 𝑎𝑏 𝑏 2 ) 𝑎3 𝑏 3Difference of cubes(𝑎 𝑏)(𝑎2 𝑎𝑏 𝑏 2 ) 𝑎3 𝑏 3Binomial cubes(𝑎 𝑏)3 𝑎3 3𝑎2 𝑏 3𝑎𝑏 2 𝑏 3(𝑎 𝑏)3 𝑎3 3𝑎2 𝑏 3𝑎𝑏 2 𝑏 3Example 5: Find the product(2𝑥 3)2Use perfect square formula(2𝑥)2 2(2𝑥)(3) (3)2Simplify each term4𝑥 2 12𝑥 9Final answerExample 6: Find the product(2𝑥 𝑦)(2𝑥 𝑦)(4𝑥 2 𝑦 2 )(4𝑥 2 𝑦 2 )(4𝑥 2 𝑦 2 )16𝑥 4 𝑦 4Multiply first two binomials withdifference of squares formulaMultiply solution by last binomial withdifference of squares formulaFinal answerFactoring PolynomialsTo factor a polynomial means to do the reverse of multiplying, that is to find an equivalentexpression that is a product. Factoring is an important algebraic skill and in this section we studythe types of factorization that will commonly arise in your study of mathematics.Terms with common factors:When factoring, you should always look for factors common to all the terms of an expression.32

Example 7: Factor4𝑦 2𝑛 8Factor out common factor of 44(𝑦 2𝑛 2)Final answerExample 8: Factor5𝑥 4𝑛 20𝑥 3𝑛Factor out common factor of 5𝑥 3𝑛5𝑥 3𝑛 (𝑥 𝑛 4)Final answerExample 9: Factor8𝑝5𝑛 𝑞 2𝑛 4𝑝4𝑛 𝑞 3𝑛 2𝑝4𝑛 𝑞 4𝑛Factor out common factor of 2𝑝4𝑛 𝑞 2𝑛2𝑝4𝑛 𝑞 2𝑛 (4𝑝𝑛 2𝑞 𝑛 𝑞 2𝑛 )Final answerFactor by groupingIn more complicated expressions, there may be a common binomial factorExample 10: Factor(𝑎 𝑏)(𝑥 7) (𝑎 𝑏)(𝑥 𝑦 2 )Factor out common binomial factor of (𝑎 𝑏)(𝑎 𝑏)[(𝑥 7) (𝑥 𝑦 2 )]Combine like terms in second factor(𝑎 𝑏)(2𝑥 7 𝑦 2 )Final answerExample 11: Factor𝑎𝑥 2 𝑎 𝑥 2 1Factor GCF out of first and second group𝑎(𝑥 2 1) 1(𝑥 2 1)Factor common binomial factor(𝑥 2 1)(𝑎 1)Factor difference of squares(𝑥 1)(𝑥 1)(𝑎 1)Final answer33

Factor difference of squaresExample 12: Factor25𝑥 2𝑛 49𝑦 2𝑛Use difference of squares formula(5𝑥 𝑛 7𝑦 𝑛 )(5𝑥 𝑛 7𝑦 𝑛 )Final answerExample 13: Factor(𝑟 3)2 (𝑠 3)2Difference of squares on groups[(𝑟 3) (𝑠 3)][(𝑟 3) (𝑠 3)]Distribute negative through parentheses(𝑟 3 𝑠 3)(𝑟 3 𝑠 3)Combine like terms in each factor(𝑟 𝑠 6)(𝑟 𝑠)Final answerExample 14: Factor81𝑥 4 (𝑦 3𝑧)2Difference of squares formula[9𝑥 2 (𝑦 3𝑧)][9𝑥 2 (𝑦 3𝑧)]Distribute negative through parentheses(9𝑥 2 𝑦 3𝑧)(9𝑥 2 𝑦 3𝑧)Final answerFactor sum and difference of two cubesExample 15: Factor𝑢3 8𝑣 3Sum of cubes formula(𝑢 2𝑣)(𝑢2 2𝑢𝑣 (2𝑣)2 )Simplify(𝑢 2𝑣)(𝑢2 2𝑢𝑣 4𝑣 2 )Final answerExample 16: Factor:3464𝑐 3 27𝑑6Identify cube roots(4𝑐)3 (3𝑑 2 )3Difference of cubes formula(4𝑐 3𝑑2 )((4𝑐)2 (4𝑐)(3𝑑 2 ) (3𝑑)2 )Simplify(4𝑐 3𝑑2 )(16𝑐 2 12𝑐𝑑 2 9𝑑 4 )Final answer

Factor trinomialsA factorization of a trinomial 𝑝𝑥 2 𝑞𝑥 𝑟 (also called quadratic expression) where 𝑝 and 𝑞 areintegers, must be of the form𝑝𝑥 2 𝑞𝑥 𝑟 (𝑎𝑥 𝑏)(𝑐𝑥 𝑑)Where 𝑎, 𝑏, 𝑐, and 𝑑 are integers. It falls that 𝑎𝑐 𝑝, 𝑏𝑑 𝑟, and 𝑎𝑑 𝑏𝑐 𝑞. Only a limitednumber of choices for 𝑎, 𝑏, 𝑐, and 𝑑 satisfy these conditions. If none of the choices work, then𝑝𝑥 2 𝑞𝑥 𝑟 is irreducible. Trying the various possibilities, as we will see in the next example,is called the method of trial and error. This method is also applicable to trinomials of the form𝑝𝑥 2 𝑞𝑥𝑦 𝑟𝑦 2 , in which case the factorization must be of the form(𝑎𝑥 𝑏𝑦)(𝑐𝑥 𝑑𝑦).Example 17: Factor6𝑥 2 7𝑥 20Identify relationshipsof (𝑎𝑥 𝑏)(𝑐𝑥 𝑑)Assume 𝑎 and 𝑐 are both positive,consider possible combinationsfor each𝑎𝑐 6𝑏𝑑 20𝑎𝑑 𝑏𝑐 7𝑎𝑐166123This gives the following possibilities326𝑥 2 7𝑥 20 (𝑥 𝑏)(6𝑥 𝑑)6𝑥 2 7𝑥 20 (6𝑥 𝑏)(𝑥 𝑑)6𝑥 2 7𝑥 20 (3𝑥 𝑏)(2𝑥 𝑑)6𝑥 2 7𝑥 20 (2𝑥 𝑏)(3𝑥 𝑑)𝑏𝑑1 20 1202 10 2104 5(3𝑥 4)(2𝑥 5) 4 5 55 4 4Next consider possible values for𝑏 and 𝑑, as 𝑏𝑑 20 we list possiblecombinationsTrying various values we find𝑏 4, 𝑑 5, 𝑎 3, and 𝑐 2Final answerAs a check, you should multiply the final factorization to see whether the original polynomial isobtained.35

Factoring with GCFExample 18: Factor 2𝑥 2 4𝑥 2𝑥 3Reorder2𝑥 3 2𝑥 2 4𝑥Factor GCF of 2𝑥2𝑥(𝑥 2 𝑥 2)Factor trinomial2𝑥(𝑥 2)(𝑥 1)Final answerExample 19: Factor2𝑥 2 20𝑥 50Factor GCF of 22(𝑥 2 10𝑥 25)Note form of perfect square trinomial2(𝑥 2 2(5)(𝑥) 52 )Perfect square formula2(𝑥 5)2Final answerExample 20: Factor63 2𝑥 𝑛 𝑥 2𝑛Product 𝑎𝑐 63( 9)(7) 63Sum 𝑏 2 9 7 2Using the 𝑎𝑐 method, split middle termSplit middle term using 9 and 763 9𝑥 𝑛 7𝑥 𝑛 𝑥 2𝑛Factor by grouping9(7 𝑥 𝑛 ) 𝑥 𝑛 (7 𝑥 𝑛 )Factor binomial GCF(7 𝑥 𝑛 )(9 𝑥 𝑛 )Final answerSolving Quadratic EquationsDefinition of a Quadratic Equation: A quadratic equat

Pre-Calculus by Abed, Farag, Lane, Wallace, and Whitney is licensed under the creative commons attribution, noncommercial, share alike 3.0 unported (CC BY-NC-SA 3.0). This means: You are free: To Share – to copy, distribute and transmit the work . To Remix – to ada