MTH 06. Basic Concepts of Mathematics II. Uma N. Iyer Department of Mathematics and Computer Science Bronx Community College

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1 MTH 06. Basic Concepts of Mathematics II Uma N. Iyer Department of Mathematics and Computer Science Bronx Community College

2 ii To my parents and teachers IthankAnthonyWeaverforeditingthisbook. Acknowledgements U. N. Iyer Copyright Uma N. Iyer 04. Department of Mathematics and Computer Science, CPH 5, Bronx Community College, 55 University Avenue, Bronx, NY 045. Version, Spring 06.

3 Contents Real Numbers. Introduction Simple exponents, roots, and absolute values Exponents and Radicals Simplifying expressions involving exponents and radicals Simplifying expressions involving exponents, radicals, and, Simplifying expressions involving exponents, radicals, and,, +, Absolute value and simplifications involving absolute values Simplifying expressions involving absolute values Homework Exercises Roots and Radicals Homework Exercises Operations on Radical expressions Homework Exercises Solving Radical Equations Homework Exercises Rational Exponents Homework Exercises Complex numbers Homework Exercises Quadratic Functions 5. Solving Quadratic Equations By Factoring Homework Exercises Completing the square and the quadratic formula Homework Exercises Introduction to Parabolas Homework Exercises Solving Word Problems Using Quadratic Equations iii

4 iv CONTENTS.4. Homework Exercises Rational Expressions 79. Introduction Homework Exercises Simplifying Rational Expressions Homework Exercises: Multiplying and Dividing Rational Functions Homework Exercises: Adding and Subtracting rational functions Homework Exercises: Complex expressions Homework Exercises: Solving Rational Equations Homework Exercises: Exponential and Logarithmic functions 0 4. Exponential functions Homework Exercises: Logarithmic functions Homework Exercises: A Polynomials 9 A. Multiplication of polynomials A.. Homework Exercises A. Division by a monomial A.. Homework Exercises A. Factoring polynomials A.. The Greatest Common Factor (GCF) A.. The Grouping method A.. The Standard Formulae A..4 Monic Quadratics in one variable A..5 Non-monic Quadratics in one variable A..6 Summary A..7 Homework Exercises

5 Chapter Real Numbers. Introduction Let us quickly recall some basic terminology. The reader is informed that some authors and teachers may use different terminology. The set of natural numbers is N {,,, }. The set of natural numbers is closed under addition. That is, given any two natural numbers, their sum is also a natural number. Addition is a commutative operation. That is, a+b b+a for any natural numbers a, b. The set N is also closed under multiplication, and multiplication is also commutative. That is, a b b a for any natural numbers a, b. Further,itcontainsthe multiplicative identity. That is, a a a foranynaturalnumbera. But N does not contain the additive identity. The set of whole numbers is W {0,,,, }. Notice that W contains just one more element than N, thenumber0. Thenumber0isthe additive identity. That is, a +0a 0+a for any natural number a. Since every natural number is also a whole number, we say that N is a subset of W, writtenmathematicallyas N W. Thinkinginnon-technicalterms,wecouldsay, N is contained in W. Notice that while W is closed under addition and multiplication, it is not closed under subtraction. For example, 7isnotawholenumber. Thisbringsustothenextnumber system. The set of integers is Z {,,,, 0,,,, }. The set Z is closed under addition, multiplication, and subtraction, and has the additive identity. Further, W is a subset of Z. SowehaveN W Z.

6 CHAPTER. REAL NUMBERS Notice that Z is not closed under division. For example, 7 isnotaninteger. Therefore,a new number system is needed which would contain Z and be closed under addition, subtraction, multiplication, and division. The set of rational numbers is denoted by Q. Itisdifficultto list all the numbers of Q. Wethereforeuseset-builder notation. Q { a b } a, b Z,b 0. such that The set of Here, is read as elements of, and is read as not equal to.. a Thus, Q is equal to the set of all fractions b such that a, b are integers, and b is not equal to 0. The set Q is closed under addition, subtraction, multiplication, anddivisionbynonzeroelements, and contains 0 (the additive identity) and (the multiplicative identity). Addition and multiplication are commutative and associative. Moreover, multiplication satisfies the distributive law over addition. That is, a (b + c) a b + a c for all rational numbers a, b, c. For an integer n, wecanviewn as a rational number as n.forexample,5 Z can be viewed as 5 Q. Therefore,N W Z Q. Herearesomemoreexamplesofrationalnumbers The rational numbers can be arranged in a line, but they leave infinitely many holes. These holes get filled by irrational numbers. Thatis,anirrationalnumbercannotbewritteninthe form of a fraction of two integers. Examples of irrational numbers are,,π (pi),e (the Euler number). Irrational numbers are extremely useful, and we will encounter them throughout our course. Numbers which are rational or irrational, are called the real numbers. The set of real numbers is denoted by R. There is a bigger number system containing R on which addition and multiplication are commutative and associative, which is closed under subtraction and division

7 .. INTRODUCTION by non-zero numbers, and which contains 0 and. That number system is the set of complex numbers denoted by C. Leti. It turns out that every element of C can be written in the form of a + bi where a, b are real numbers. That is, C {a + bi a, b R}. We will see more on complex numbers towards the end of this course. So we will focus only on the real numbers. The following figure is a Venn diagram representation of the number systems introduced so far. R Q C Z W N Irrationals In your previous courses you learnt how to add, subtract, multiply, and divide various kinds of numbers. Here we start with the notion of exponents, roots, and absolute values.

8 4 CHAPTER. REAL NUMBERS. Simple exponents, roots, and absolute values.. Exponents and Radicals Exponents are used to describe several self-multiplications. Instead ofwriting we write 5.Foranynaturalnumbern, andanyrealnumbera, wewrite a n a a a; here, a appears n times. Further, a 0 for a 0, and 0 0 is undefined. We say that a is the base and n is the exponent. Some examples: 4 :Here,thebaseisandtheexponentis4.Thevalueof 4 6. :Here,thebaseisandtheexponentis.Thevalueof :Here,thebaseis4andtheexponentis0.Thevalueof4 0. ( 5) :Here,thebaseis 5 andtheexponentis. Thevalueof( 5) ( 5) ( 5) 5. 5 :Here,thebaseis5andtheexponentis.Thevalueof ( ) ( ) :Here,thebaseis and the exponent is. 5 5 ( ) ( ) ( ) ( ) The value of The inverse operation of taking exponents is the operation of extracting aroot.thatis, 9 because because 5 5. because. 00 because because 8. 7 because because because. 00 because because ( ) 8. 7 because ( ) 7. The symbol is called the radical. In an expression a,thenumbera is called the radicand. a is read a -squared; a is read square-root of a a is read a -cubed; a is read cube-root of a

9 .. SIMPLE EXPONENTS, ROOTS, AND ABSOLUTE VALUES 5 Properties of exponents and radicals: a n a m a n+m.forexample, 4 ( ) ( ) 6. (a n ) m a nm.forexample,(5 ) (5 ) (5 ) (5 )(5 5) (5 5) (5 5) 5 6. (a b) n a n b n.forexample,(4 5) ( a ) n a n ( ) 4 for b 0. Forexample, 4 b bn a b a b for a, b 0. For example, a a 4 b for a, b 0,b 0. Forexample, b a b a b.forexample, a b a b for b 0. Forexample, Remark : (a + b) a + b.checkwithanexample.leta andb. Then Therefore, ( + ) +. (a + b) a + b (+) (a b) a b.checkwithanexample.leta andb. Then Therefore, ( ). (a b) a b ( ) ( ). 4. a + b a + b.checkwithanexample.leta andb 4. Then a + b a + b

10 6 CHAPTER. REAL NUMBERS Therefore, + +. You can check that a b a b You can check that a + b a + b You can check that a b a b.. Simplifying expressions involving exponents and radicals Now we are ready to simplify expressions involving exponents and radicals. Example : Simplify ( ) 4. Example : Simplify ( ) 5. ( ) 4 ( )( )( )( ) 8 There are five negatives. ( ) 5 ( )( )( )( )( ). Example : Simplify ( ) 98.Note,( ) 98 ( )( ) ( ) +. }{{} 98 times Example 4: Simplify 49. Note, Example 5: Simplify 98. Note, Example 6: Simplify Example 7: Simplify ( ). ( ) 9. Example 8: Simplify ( ( 5) ).Startsimplifyingfromtheinnermostparentheses. ( ( 5) ) ( 5 ) (5) 5. Example 9: Simplify ( 5 ). ( 5 ) 5. This is the meaning of 5. Yet another way of understanding the above problem is ( 5 ) ( ) ) Example 0: Simplify.Note,( 7 7 ( )

11 .. SIMPLE EXPONENTS, ROOTS, AND ABSOLUTE VALUES 7 Classroom Exercises : Find the value of (a) ( ) 0 (b) ( ) 6 (c) ( ) 4 (d) ( 8 ) (e) ( ) (f) 00 (g) ( 7 ).. Simplifying expressions involving exponents, radicals, and, Suppose we want to simplify the expression, 5. This is equal to In other words, when we encounter multiplication and exponents in the same expression, the exponents get performed first. This motivates the following rule: To simplify an expression involving exponents, radicals, multiplications, and divisions, first perform exponents and radicals, and then perform multiplications and divisions. Example : Simplify 9 (4) 5. Example : Simplify 7 44 ( ) 9 (4) ( ) 6 ( 8) 6 ( 8) 6 48 ( 8) Example : Simplify 5.(Thatis,rationalize the denominator. This means, the denominator should be free of the radical symbol.) ; recall that 5 55.

12 8 CHAPTER. REAL NUMBERS Example 4: Simplify ( 7) 7 5. Simplify radicals ( 7) 7 ( 7) Multiplication and division now Rationalize the denominator 5 5. The final answer. 6 As a rule we always rationalize the denominator to obtain the final answer. Example 5: Simplify Simplify exponents and radicals Division is multiplication by reciprocal Rationalize the denominator The final answer. 5 Many a time, one has to simplify the radicand (in a radical expression) or the base (in an exponential expression) first.

13 .. SIMPLE EXPONENTS, ROOTS, AND ABSOLUTE VALUES 9 Example 6: Simplify Example 7: Simplify (90 9 ) ( 90 (90 9 ) 9 ) Classroom Exercises : Simplify the following (Do not forget to rationalize the denominator whenever necessary) :. (a). (a). (a) 4. (a) 8 (b) 8 (c) (d) 50 (b) 7 (c) 48 (d) 75 0 (b) 45 (c) 80 (d) 5 4 (b) 54 (c) 8 (d) 6 5. (a) 8 (b) 8 (c) (d) (a) (b) 7 (c) 48 (d) (a) 0 (b) 45 (c) 80 (d) 5 8. (a) 4 (b) 54 (c) 8 (d) ( 5 )

14 0 CHAPTER. REAL NUMBERS (7 5 ) 7. (4 4 ) 0..4 Simplifying expressions involving exponents, radicals, and,, +, Recall, to simplify an expression involving,, +,, wefirstperform, (left to right) and then perform +, (left to right). Therefore, to simplify an expression involving exponents, radicals,,, +, and we perform the operations in the following order:. Exponents and Radicals are computed first.. Multiplications and Divisions are computed second (left to right).. Additions and Subtractions come at the end (left to right). Example : Simplify Exponents and radicals first Multiplications and divisions next (left to right) Example : Simplify The least common denominator is Simplify exponents and radicals first Multiplication is commutative Additions and subtractions last.

15 .. SIMPLE EXPONENTS, ROOTS, AND ABSOLUTE VALUES Example : Simplify ( ) +( 9). ( ) +( 9) Notice in this case that Classroom Exercises : Simplify (a) (b) (c) (d) 5 +4 (e) (f) Absolute value and simplifications involving absolute values The absolute value of a real number is its distance from 0 on the real number line. Absolute value of number a is denoted by a The distance between 4 and 0 on the number line is 4. Hence, 4 4. The distance between 4 and0onthenumberlineis4.hence, 4 4. The distance between 0 and 0 on the number line is 0. Hence, 0 0. The absolute value is closely related to exponents and radicals. Pay careful attention to the numbers in the following three examples ( 4) In other words, a a for any real number a.

16 CHAPTER. REAL NUMBERS Classroom Exercises : (a) 5 (b) ( 5) (c) 7 (d) ( 7) (e) 7 (f) Mind you, a a. Forinstance, ( ) 8 whichisnotthesameas. Properties of the absolute value: a b a b for any real numbers a, b. Forexample, ( ) 6 6. a a for any real numbers a, b with b 0. Forexample, b b. Remark: a + b a + b. Checkfora ( ) and b. ( ) +. Whereas, + +. a b a b. Checkfora ( ) and b. ( ). Whereas,. Classroom Exercises : Find the values of (a) 9 0 (b) ( 7) (c) (d) ( 9 0 ) ( ( 7))

17 .. SIMPLE EXPONENTS, ROOTS, AND ABSOLUTE VALUES..6 Simplifying expressions involving absolute values When we want to simplify an expression involving exponents, radicals,,,, +,, theabsolute value takes the same priority as exponents and radicals. That is,. Exponents, Radicals, and Absolute Values are computed first.. Multiplications and Divisions are computed second.. Additions and Subtractions come at the end. As always, if exponents, radicals, and absolute values themselves have some complicated expressions within them, then the inner complications have to be simplified. In other words, exponents, radicals, and absolute values can themselves be grouping mechanisms. Example : Simplify Absolute value, exponent and radical first 0 9 Multiplication next Subtraction last. Example : Simplify Exponent, radical, absolute value first Multiplication and division next Addition last; the least common denominator is

18 4 CHAPTER. REAL NUMBERS Example : Simplify Inthiscase, theoutermostabsolute value is to be performed after the expression inside is simplified () Classroom Exercises: (a) 4 (b) 4 (c) 4 (d) 8 (e) ( ) 8 (f) ( ) 6 (g) ( ) (h) ( ) (i) 7 ( ) ( ) The inner exponents, radicals and absolute values are to be simplifed first Simplify multiplications and divisions inside

19 .. SIMPLE EXPONENTS, ROOTS, AND ABSOLUTE VALUES 5..7 Homework Exercises Do not forget to rationalize the denominator whenever necessary. Simplify (exponents and radicals):. ( ) 00. ( 5). ( ( ) 4 ) ( 5 ) ( 7 ) Simplify (exponents, radicals, multiplications, and divisions) ( )

20 6 CHAPTER. REAL NUMBERS 0. ( ( 5) ( 4) ) Simplify (exponents, radicals, multiplications, divisions, additions, and subtractions) ( ) ( 4) 9+( ) Simplify (absolute values, exponents, radicals, multiplications, divisions, additions, and subtractions) 8. ( ) ( 7) 0. ( ( ) 4 ) ( ) ( ) )

21 .. ROOTS AND RADICALS ( 4) Roots and Radicals We were introduced to roots and radicals in Section.4. Recall that the symbol is called the radical and the expression inside the symbol is called the radicand. For example, xy xy in the expression, the radicand is a + b a + b. For a natural number n, wesaythata is an n-th root of a number b if and only if a n b. We write the phrase, n-th root of b, in short as n b.therefore, a n b if and only if a n b. ( ) The first thing to note is that by definition, n n b b. Terminology : The second root of b is called the square-root of b; itisdenotedby b instead of b.thethirdrootofbis called the cube-root or the cubic-root of b. First we note that there are two square-roots of 5 as 5 5and( 5) 5. Similarly, as 4 8,and( ) 4 8,therearetwofourthrootsof8. Theprincipal square-root of anon-negativenumberisitspositive square-root. Similarly, the principal fourth root of a non-negative number is its positive fourth root. For example, the principal square-root of 9 is +. We therefore write 5 5, 5 5; 4 8, 4 8. In general, for an even natural number n, n, the principal n-th root is the positive n-th root. Note that, the n-th root of a real number need not always be a real number. For example, 9isnotarealnumber. Thisisbecause,thesquareofarealnumber is non-negative. For instance, 9, and ( ) 9. In other words, there is no real number a, suchthata 9. Therefore, 9isnotareal number. In fact, 9isacomplexnumber,andwewilllearnaboutcomplexnumbersin Section 6..

22 8 CHAPTER. REAL NUMBERS Likewise, 4 65 is not a real number because the fourth power of any real number is nonnegative. Again, 4 65 is a complex number. Note, , and ( 5) You probably have realized the following generalization: For n an even natural number, then-th root of a negative real number is not a real number. This is not to be confused with the following situation: What is square-root of? It is the irrational number. There is no simpler way of writing. This is a real number. Likewise, 0, 4 5, or 9 00 are all irrational real numbers. Here are some examples. n 0 0 because 0 n 0for any natural number n because is not a real number because the square of a real number can not be negative because because ( 4) ( 4) ( 4) ( 4) because is not a real number because the fourth power of a real number cannot be negative. 5 because 5. 5 because ( ) 5 ( ) ( ) ( ) ( ) ( ). 7 ( ) because because ( 0 ) ( 0 ) ( 0 ) ( ) is not a real number because the square of a real number can not be negative.

23 .. ROOTS AND RADICALS 9 Here are some important properties of radicals : Property For real numbers a, b and natural number n, if n a, n b are real numbers, then n a b n a n b. Proof : Suppose x n a.thenx n a. Suposey n b.theny n b. Note that }{{} x n y n (x y) n. }{{} a b In other words, a b (x y) n.hence n a b x y. Thatis, n }{{ a } b }{{} x y. }{{} n n a b a n b We have therefore proved that n a b n a n b. For example : (This can not be simplified further.) (This can not be simplified further.) (This can not be simplified further.) n Property For real numbers a, b, withb 0,andanaturalnumbern such that a, n b are real numbers, then n a n a b n. b Proof Suppose x n a.thenx n a. Suposey n b.theny n b. Note that a ( ) b xn x n y n. y That is, a b ( x y ) n.

24 0 CHAPTER. REAL NUMBERS n a Therefore, b x y. n a n a That is, b n. b For example : Here, we need to rationalize the denominator as This can not be simplified further Here,weneedtorationalizethedenominator This can not be simplified further. Caution : n a + b n a + n b and n a b n a n b. For example, 4+9, while Similarly, , while Recall from the definition and properties of the absolute value of a real number. The absolute value of a real number is its distance from 0 on the number line. The absolute value of x is denoted by x. So,,and. Thisisbecause,bothand areatadistanceof from0onthenumberline.notethat 9,and ( ) 9. Ingeneral, x x for any real number x.

25 .. ROOTS AND RADICALS Here is yet another situation: ( 5) In general, 4 x 4 x for any real number x. One more example: ( 0) In general, 6 x 6 x for any real number x. Note that for any even natural number n, andanyrealnumberx, wegetx n to be a nonnegative number. Therefore, the principal n-th root of x n will be a non-negative real number. Therefore, For an even natural number n, andanyrealnumberx, wehave n x n x. Now consider the following situation: 8. ( ) 8. In general, x x for any real number x. Here is yet another situation: ( ) In general, 5 x 5 x for any real number x. We can draw a general conclusion from this: For an odd natural number n, andanyrealnumberx, wehave n x n x. Here are some examples on simplifying algebraic expressions. Assume that all the variables are positive. 8x 9x. Thiscannotbesimplifiedfurther. 80x 6 5 x x 6 x 5 x 6 x 5 x 4x 5x.

26 CHAPTER. REAL NUMBERS x y x 4x x x x. y y y Now rationalize the denominator x x y x x y y y x xy. y There are several ways of arriving at this simplification. Here is another one: x y 40x4 40x 4 y 5 y 5 x y y y 40x 4 y 5 x y y 8x 5x y y Here, we need to rationalize the denominator. x 5x y x 5x y y y y x 5xy y y y 4x xy y 8x 5x y y x 5xy y y Here is another way of arriving at the same simplification: 40x4 y 5 40x4 y y 5 y 40x4 y 40x 4 y y 6 y 6 40x 4 y y 6 x xy. y x 5x y y. x 5xy y. 8x 5xy y 6 x 5xy y. Pythagorean Theorem: Given a right angled triangle, the sum of the squares of the lenghts of the legs is equal to the square of the the length of the hypotenuse. That is, a c Pythagorean Theorem: a + b c b

27 .. ROOTS AND RADICALS x Example: Find the value of x (simplify the radical). (x) + x 5 By Pythagorean theorem 4x + x 5 5x 5 Divide both sides by 5 5 x 45 Take square-roots of both sides x 45 5units We only consider the positve square-root. x An important application of the Pythagorean theorem is the distance formula. Y (x,y ) Given two points (x,y )and(x,y )onthecoordinateplane, let the distance between them be d. By Pythagorean theorem we have d (x x ) +(y y ). d y y Thus, we have the distance formula (x,y ) x x X d (x x ) +(y y ) For example, to find the distance between points (, ) and (, 4), we first set (x,y )(, ) and (x,y )(, 4). Then, the distance d is given by d ( ( )) +( 4) ( + ) +( ) 5 +( ) Classroom Exercises :. Simplify (rationalize the denominator if needed): (a) (b) (c) (d) (e) ( 4) 4 (f) 8 + (g) + 7 (h) (i) 64 y 7 (j) z 8 (k) 4 4 (l) 4 ( ) 4 (m) (n) y 7 ( ) (o) z 6

28 4 CHAPTER. REAL NUMBERS. Find x in the following right angled triangles (simplify the radicals): 7 x 4 7 x x x 6 x x x 8 x x x. Find the distance between the given pair of points: (a) (, 4) and (, 4). (b) (, ) and (5, 6). (c) (, 4) and (, 4). (d) (, 6) and (0, 6)... Homework Exercises. Simplify and rationalize the denominator whenever necessary. The variables here take positive values only. (a) 6 ( ) 6 (b) ( ) (c) (d) 75

29 .. ROOTS AND RADICALS 5 (e) 4 4 (f) (g) (h) 8 (i) (j) (k) (l) (m) ( 6 0 ) 6 6 (n) ( 0) 6 8 (o) x 8 7 (p) x 7 5 x 5 (q) y x 7 (r) y 9 (s) (t) 7x 5 y z 75a 48a 6 4 b 8

30 6 CHAPTER. REAL NUMBERS. Find x in the following right angled triangles (simplify the radicals): x 4 0 x x x x 8 x 9 x 5 x x x. Find the distance between the following pairs of points on the coordinate plane. (a) (, 8) and ( 9, 7) (b) (, 0) and (0, ) (c) (, 5) and (9, 7) (d) (, 4) and (, ).4 Operations on Radical expressions In this lesson we continue working with radical expressions following the properties presented in lesson 8. Recall For a natural ( ) number n, andrealnumbersa, b, wesaya n b if and only if a n b. That is, n n b b. On the other hand, n x n x if n is an even natural number, and x is any real number. If x is allowed only positive values, then n x n x. If n is an odd natural number, then n x n x for any real number x.

31 .4. OPERATIONS ON RADICAL EXPRESSIONS 7 For n an even natural number, and a negative, n a is not a real number. But if n is an odd natural number, then n a is a real number for any real number a. For n any natural number, and a, b any real numbers such that n a, n b are real numbers, we have n a b n a n b. For n any natural number, and a, b any real numbers such that b 0,and n a, n b are real numbers, we have n a b n a n b. Caution : n a + b n a + n b and n a b n a n b. Now we can proceed with some examples of operations on radical expressions. A strategy for simplifying radicals is to look for perfect n-th power factors of n (Here, we treat terms involving asliketerms. Forinstance, x +5x 8x.) (We treat terms involving 5asliketerms. Forinstance,9x x x.) Here we cannot proceed unless we simplify the individual radical terms This cannot be simplified further as 4 5and 4 9areunlikeradicalexpressions This cannot be simplified further as 4 5and 5areunlikeradicalexpressions.

32 8 CHAPTER. REAL NUMBERS Thisneedscarefulanalysis The least common denominator is Wefirstsimplifytheradicalexpressions,andrationalizethe denominator The least common denominator is

33 .4. OPERATIONS ON RADICAL EXPRESSIONS x 7x 6x.Again,firstrationalizethedenominator. 4x 7x 6x 4x 7x 4x 6x 4x 4x 7x 4x 64x 4x 7 x 4 4 4x 4 4x 4 4 x 4x 7 4x 4 7 4x 4 4 4x 7 4x 4 4x We have two distinct kinds of radical expressions. So, combining like radicals, which cannot be simplified further This cannot be simplified any further. Note that the cube-root and the fourth root are distinct kinds of roots This cannot be simplified any further. Note that the cube-root and the fourth root are distinct kinds of roots. 4 ( 5 ).Multiplicationdistributesoversubtraction.So, 4 ( )

34 0 CHAPTER. REAL NUMBERS 5. ( x y ).Assumethatthevariablesarenon-negative. ( x y ) ( x y )( x y ) x x x y y x + y y x xy yx + y x xy xy + y x xy + y 6. Note that ( x) ( y ) x y. Thus,( x) ( y ) ( x y ). 7. ( x + y ).Assumethatthevariablesarenon-negative. ( x + y ) ( x + y )( x + y ) x x + x y + y x + y y x + xy + yx + y x + xy + xy + y x + xy + y 8. Note that ( x) + ( y ) x + y. Thus,( x) + ( y ) ( x + y ). 9. ( x y )( x + y ). Assume that the variables are non-negative. We follow the rules of polynomial multiplication. ( x y )( x + y ) x x + x y y x y y x + xy yx y x + xy xy y x y 0. ( x + y ) ( x x y + y ) Assume that the variables are non-negative. ( x + y ) ( x x y + y ) x x x x y + x y + y x y x y + y y x x y + xy + yx yxy + y x x y + xy + x y xy + y x + y.. ( x y ) ( x + x y + y ) Assume that the variables are non-negative. ( x y ) ( x + x y + y ) x x + x x y + x y y x y x y y y x + x y + xy yx yxy y x + x y + xy x y xy y x y.

35 .4. OPERATIONS ON RADICAL EXPRESSIONS The examples 6, 7, and 8 are the radical versions of certain algebraicformulae. Recalland compare the formulae below. Name Recall Compare Difference of squares (a b)(a + b) a b ( )( ) x y x + y x y Sum of cubes (a + b)(a ab + b ( ) x + y ) ( x x y + ) y a + b x + y Difference of cubes (a b)(a + ab + b ( ) x y ) ( x + x y + ) y a b x y These formulae help us to rationalize the denominators of certain complicated radical expressions. Here are some examples:. 5 ( + 5 ) ( 5 ) ( + 5 ) ( + 5 ) 5 ( + 5 ) ( ) ( ) ( 7 9 ) ( ) 4 ( ) ( ) ( )

36 CHAPTER. REAL NUMBERS. Here is a general case, which will be relevant when we work with complex numbers: a + b c (a b c) (a + b c) (a b c) a b c a (b c) Classroom Exercises : Simplify (a) 5+ 5 (b) (c) (d) (e) (f) (g) (h) 5( 5+ ) (i) ( 8 ) (j) ( 5 + ) a b c a b c. (k) (l) (m) ( ) (n) ( + 5) (o) ( 7 ) (p) ( ) (q) ( + )( ) (r) ( 7 5)( 7+ 5)

37 .4. OPERATIONS ON RADICAL EXPRESSIONS (s) ( 5+ 7 )( 5 7 ) (t) 5 6 (u) (v) 4m m m Rationalize the denominators : (a) (b) + (c) Homework Exercises Perform the following operations. () () () (4) (5) ( ) (6) ( ) (7) 5 ( ) ( + ) ( ) ( )( ) 5 (8) ( + 5 )( ) ( + 5 )( 5 ) ( )( ) (9) ( 5 )( + 5+ ) Rationalize the denominators. ;. 4 ;. 6 ; 4 ; 4 ; 4 7 ; ; ; + 7.

38 4 CHAPTER. REAL NUMBERS.5 Solving Radical Equations Radical equations are equations involving radical expressions. In this lesson we will learn to solve certain simple radical equations. Remember to check that your solutions are correct. Example : Solve x forx. x ( x) Square both sides x 9. This is a possible solution. Checking whether the solution is correct: 9. We see that x 9is a solution. Example : Solve x 5forx. x 5 5 x ( ( ) 5 x Divide both sides by ; ) Square both sides; x 5 4 x 5 4 Divide both sides by x 5. This is a possible solution. Checking whether the solution is correct: We see that x Example : Solve x 45forx. is a solution. x 45 5 x 4 ( ( ) 5 x 4 Divide both sides by ; ) Square both sides; x x 5 +4 Add 4 to both sides; 4

39 .5. SOLVING RADICAL EQUATIONS 5 x Simplifying the right hand side; x 4 4 Divide both sides by ; x 4. This is a possible solution. Checking whether the solution is correct: We see that x is a solution. Example 4: Solve 4 x forx. 4 x x Divide both sides by 4; 4( ( ) x 4) Square both sides; x 6 x + Add to both sides; 6 x Simplifying the right hand side; x 6 Divide both sides by ; x. 6 This is a possible solution. Checking whether the solution is correct: We see that x is not asolution. 4 6 There is no solution to the given equation. Example 5: Solve x 5+7forx. x 5+57 x 5 Subtract 5 from both sides; x 5 Divide both sides by ;

40 6 CHAPTER. REAL NUMBERS ( x 5 ) ( ) Square both sides; x x x Add 5 to both sides; Simplifying the right hand side; x 49 Divide both sides by ; 9 x 49. This is a possible solution. 8 Checking whether the solution is correct: We see that x 6 Example 6: Solve x +x for x. x +x is a solution. ( x + ) x Square both sides; x +x 0x x Subtract x and subtract from both sides; 0(x )(x +) Factoring the right hand side; (x ) 0 or (x +)0 Solving for x x or x Checking whether the solutions are correct: These are possible solutions. So, x is a solution. ( ) + + So, x is not a solution. Example 7: Solve 0(x +)x +forx. 0(x +)(x +) Place parentheses around the right hand side; ( 0(x +) ) (x +) Square both sides;

41 .5. SOLVING RADICAL EQUATIONS 7 0(x +)x +6x +9 Multiply the right hand side; 0x +0x +6x +9 0x +6x +9 0x 0 Subtract 0x and subtract 0 from both sides; 0x 4x Simplify the right hand side; 0(x 7)(x +) Factoring the right hand side; (x 7) 0 or (x +)0 Solving for x x 7 or x These are possible solutions.; Checking whether the solutions are correct: 0(7 + ) ( +) So, x 7is a solution. So, x is a solution. Example 8: Solve x + x +80 x + x +80 x x +8 Place the two radical expressions on opposite sides of ; ( x ) ( x +8 ) Square both sides; x +x +8 0x +8 x Subtract x and from both sides; 0x +5 5 x Subtract 5 from both sides. Checking whether the solutions are correct: 5 + ( 5) These are not real numbers. There is no solution to this problem. Example 9: Sometimes you will need to square twice. Solve the equation x +7+ x + for x. x +7+ x + First isolate one of the radicals;

42 8 CHAPTER. REAL NUMBERS x +7 x + Subtract x + from both sides; ( x +7 ) ( x + ) Square both sides; x x ++x + x +7x +7 4 x + x +7 x 7 4 x + x 4 x + (x) ( 4 x + ) Multiplying and simplifying the right hand side; Further simplifying the right hand side; Subtracting x and 7 from both sides; Simplifying the left hand side; Square both sides; 4x 6(x +) 4x 6x +48 4x 6x 48 0 Subtracting 6x and 48 from both sides; 4(x 6)(x +)0 Factoring the left hand side; x 6orx These are the possible solutions. Checking whether the solutions are correct: (6) ( ) Example 0: Solve the literal equation x +7y for x. x +7)y x 6is not a solution x is a solution ( x ) +7 y Square both sides; x +7y x y 7 x y 7 y x ± 7 y 7 x ± Subtract 7 from both sides; Divide both sides by. Take square-roots of both sides; Note the two options ±. Wehavesolvedforthevariablex.

43 .5. SOLVING RADICAL EQUATIONS 9 Classroom Exercises: Solve the following equations for x.. x 4 x 7 x 5. x + x 4 x. x +5 x 7 4 x x +5 5 x 7+8 5x x +6 x x +x + 5 x +x 6. x x +70 x +6 x +0 x +5 x x x +7 x +6 x + x +5 x + Solve the following literal equations for the indicated variables. c (a ) +(b 4) for b. B A for C. C y x +4forx..5. Homework Exercises Solve the following equations for x.. x 8 x 5 x. x +4 x 46 x x + 4 x 5 x 5 4. x 5 4 5x ++ x x +7 x + 5x + 7 x x x x + x 40 x ++ x 0 x +4 x 0 7. x + x 4 x ++ x x +4 x Solve the following literal equations for the indicated variables. c (a ) +(b 4) for a. B A for B. C y x 4forx.

44 40 CHAPTER. REAL NUMBERS.6 Rational Exponents Recall from subsection.. and section. the notion of n-th root. For n> an integer, we say that n b a if and only if a n b. We introduce rational exponents for the ease of computations. For a real number b and any integer n>, we set b n n b. Notice that if n is even, then the n-th root of a negative number is a complex number. We therefore assume that b is positive whenever n is even. Examples: ( ) Classroom Exercises: Evaluate (a) 5 (b) 65 4 (c) 4 (d) 8 7 (e) 5 (f) 65 4 (g) ( 4) (h) 8 7 One immediate consequence of this notation is that for any real number a and integers m, n with n>, we have a m n n a m ( n a ) m. Examples: 4 ( 4 ) 8 64 ( 64 ) 4 6 ( ) 5 ( 5 ) ( ) 4 ( 4 ) 8 64 ( 64 ) ( 5 ) ( 5 ) ( 4 8 ) ( 4 8 ) 9. Classroom Exercises: Evaluate (a) 5 (b) 65 4 (c) 4 (d) 8 7 (e) 5 (f) 65 4 (g) ( 4) (h) We can now refer to the properties described in subsection.. and section. and state them as follows: For any real numbers a, b and rational numbers m, n a m a n a m+n, (ab) m a m b m (a m ) n a mn, a m a m for a 0. a m a n am n for a 0; ( a ) m a m b b m for b 0;

45 .6. RATIONAL EXPONENTS 4 Examples: x 7 x 5 7 x x 8 7 y 5 y 4 y y y x x ( 5 ) x 5 x x 4 5 b 5 6 b b 5 6 b 0 b 9 b 4 (5 8) 5 8 ( 5 ) ( 8 ) (8 6) (x y) 7 x 7 y 7 (p q) 4 5 p 4 5 q 4 5 ( ) ( 5 5 ) ( ) ( ) ( ) x 7 x ( ) 4 7 p 5 p 4 5 y y 7 q q 4 5 ( ) 5 0 ( ) ( 5 ) ( ) x x 4 5 x 8 5 x 7 Classroom Exercises: Evaluate (a) (b) (c) (49 5) (4 ) 5 (5 000) ( ) ( ) ( (d) 7 ) ( ) (e) 5 8 ( ) ( ) (f) 7 64 ( 5) x 7 7

46 4 CHAPTER. REAL NUMBERS Classroom Exercises: Simplify and write your results with positive exponents: (g) (x ) x y 5 (y ) 4 (h) (4x y ) (x 5 y 4 ) 4.6. Homework Exercises Evaluate z 5 z z ( x 4 y z () () ( 5) 5 () (4) ( 5) 4 5 Evaluate (5) (6) ) (7) (8 5) (4 ) 5 (5 000) ( ) ( ) ( (8) 64 ) ( ) (9) 7 8 ( ) ( ) (0) ( 7) Simplify and write your results with positive exponents: () (x ) 4 x y 5 (y ) 5 () (4x y ) (x 5 y 4 ) 4.7 Complex numbers z 5 z z ( x y z 5 Recall the argument why 9. Note, 9. Therefore 9. Nowrecalltheargument why 9cannotbearealnumber. Thesquareofarealnumbercannotbenegative. Therefore, there is no real number which is 9. )

47 .7. COMPLEX NUMBERS 4 Complex numbers allow us to take square-roots of any real number. We first define i. Therefore, i ( ). The letter i standsfor imaginary. Every complex number z can be written in the form of z a+bi where a, b are real numbers. The real number a is called the real part of z, andtherealnumberb is called the imaginary part of z. Forinstance,therealpartofthecompexnumber+4i ( is, and its imaginary part is 4. Similarly, the real part of 7 ) 9 i is and its imaginary part is 7 9. Complex numbers are arranged on a plane. The horizontal axis is the real number line and the vertical axis is the imaginary line. These two lines intersect at the complex number 00+0i. A complex number is then plotted on this plane. The complex number a + bi is positioned at the point (a, b) asincoordinategeometry.herearesomeexamples: Imaginary axis i +i +i i i 4 0 Real axis 4 i i i i Power of i i i i i 4 Value i i i i i i i i Im i 0 i 4 Re i i

48 44 CHAPTER. REAL NUMBERS This table allows us to find other powers of i. i i 8+ i 8 i i i. Notice that we write 8 + because 8 is a multiple of 4. In other words, we separate out as many 4 s as possible from the exponent first. Mathematically speaking, we divide by 4, and get a remainder of. That is, 4 +. Geometrically, this canbeseen bygoing around0counter-clockwise starting from in the above diagram, and reading, i i, i,i i, i 4,i 5 i, i 6,i 7 i, i 8,i 9 i, i 0,i i. i 45 i 44+ i 44 i i i. Here, dividing 45 by 4 gives us a remainder of. Check: i 98 i 96+ i 96 i. Again, dividing 98 by 4 gives us a remainder of. Check: i i i4 i i. Geometrically, this is obtained by going clockwise around 0 i starting from, and reading, i i. i i i4 i i. i i i4 i i. i 7 i 7 i 4+ i 4 i i i4 i i i 54 i 54 i 5+ i 5 i. Check: When 54 is divided by 4, the remainder is. We get Addition and subtraction of complex numbers follow the same rules as radical expressions. This is expected because i is the radical expression. Here are some examples:. ( + i)+(5+4i) (+5)+(i +4i) 7+7i.. ( + i) (5 + 4i) +i 5 4i ( 5) + (i 4i) i.. ( + 5i)+(+5i) ( +)+( 5+5)i 4. ( + 5i) ( + 5i) + 5i 5i ( ) + ( 5 5)i

49 .7. COMPLEX NUMBERS ( ) 5 7i + (+ ) i 5 7i ++ i ( ) ( 7i + ) i ( ) ( 7 + i + ) 5 i ( ) ( + 5 i + ) i ( ) ( ) i i Combine the real parts and the imaginary parts Get common denominators 6. ( ) 5 7i (+ ) i 5 7i i Distribute the sign ( ) 5 + ( 7i ) i Combine the real parts and the imaginary parts ( 5 5 ) ( 7 + i ) 5 i Get common denominators ( 5 0 ) ( + 5 i ) i ( ) ( ) 0 + i i Multiplication of complex numbers also follow the same rules as for multiplication of radical expressions. Keep in mind that i. Because of this, the product of two complex numbers can be written in the form of a + bi, withnohigherpowerofi appearing. Here are some examples: (4 + 5i) +5i ( ) i(4 5i) i 5 i i +55+i. 4 9i i 6i 6.

50 46 CHAPTER. REAL NUMBERS 5 5i i 55i 55. ( + i)(5 + 4i). ( + i)(5 + 4i) (5+4i)+i(5 + 4i) Distribution ( + 5i)( + 5i). ( ) 0+8i +5i + }{{ i } 0+8i +5i i is equal to ( ) (0 ) + (8i +5i) Combining real and imaginary parts +i. ( + 5i)( + 5i) ( + 5i)+ 5i( + 5i) Distribute ( )(+ 5 7i ) i ( )(+ 5 7i ) i (+ ) 5 i +5 i + 5i +5 ( ) 5 i +5 i + 5i 5 5 i is equal to ( ) ( 5 5) + (5 + 5)i This cannot be simplified further. 7i (+ ) i i 7 i 7 ( ) i i 4 i + 7 ( ) ( ( ) + 5 ( ) + 5 ) i ( 5 ( i i ) i ) i Distribute i is equal to ( ) Collect the real and imaginary parts Get common denominators

51 .7. COMPLEX NUMBERS 47 Before we can divide complex numbers we need to know the concepts of the complex conjugate and the norm of a complex number. The complex conjugate of the complex number a + bi is a bi. Forexample The complex conjugate of + 4i is 4i The complex conjugate of 5 7i is 5 ( 7i) 5+7i. The complex conjugate of 7i is + 7i. The complex conjugate of 4 + i is 4 i. The complex conjugate of 0 is 0. Note, 0 0+0i and therefore, its complex conjugate is 0 0i which is 0. The complex conjugate of 0i is 0i. Note, 0i 0+0i and therefore, its complex conjugate is 0 0i which is 0i. The norm of the complex number a + bi is (a + bi)(a bi) a + b.thenormofa + bi is denoted by a+bi. Thenormisthethedistanceofthecomplexnumberfrom0inthecomplex plane. Examples: +4i ( ) ( + 4i)( 4i) 9 i +i 6 i i ( ) (5 7i)(5 + 7i) 5 + 5i 5i 49 i i ( 7i)( + 7i) ( ) + 7i 7i ( ( ) 7) i + 49 ( 4 + i 4 + )( i 4 ) i i + 4 i ( ) i

52 48 CHAPTER. REAL NUMBERS i ( ) 0i 0i 00 i Before we learn to divide complex numbers, recall how we rationalized the denominator of a + b c.nowwearereadytodividecomplexnumbers: (4 + 5i) (4 + 5i) (4 + 5i) (4 5i) (4 + 5i) (4 5i) 5i ( ) 6 0i +0i 5 i 5i i i Multiply the numerator and denominator by the complex conjugate of the denominator Multiply the complex numbers i (4 5i) i i (4 5i) (4 5i) i (4 + 5i) (4 5i) (4 + 5i) ( ) i +5 i ( ) 6 + 0i 0i 5 i 5 + i i i Multiply the numerator and denominator by the complex conjugate of the denominator Multiply the complex numbers

53 .7. COMPLEX NUMBERS 49 ( + 5i) ( 7+5i) ( + 5i) ( 7+5i) ( + i) (5 + 4i) +i (5 + 4i) ( + i) (5 4i) (5 + 4i) (5 4i) ( + 5i) ( 7 5i) ( 7+5i) ( 7 5i) ( ) 0 8i +5i i ( ) 5 0i +0i 6 i 0 + 8i +5i i i Multiply the numerator and denominator by the complex conjugate of the denominator 7 5 i + 5 7i 5 ( ) 5 i Multiply the complex numbers ( ) i +5 7i 5 i ( )i ( )i ( ) i ( ) 5 7i (+ ) i ( ) 5 7i (+ ) Multiply the numerator and denominator i ( ) 5 7i ( ) i (+ ) i ( ) by the complex conjugate of the denominator i

54 50 CHAPTER. REAL NUMBERS 5 5 i 7 i + 7 i 4 i + i ( ) i ( ) i ( i 7 9 ( i + ( ) i ) 4 5 i 5 ) i i 7 9 ( ) i Multiply the complex numbers Classroom Exercises: (a) Write the real and imaginary parts of + i; 4+ i; 7i; 4 8i; 0; ; i. 7 (b) What is the complex conjugate of each of the following? 7; 7i; i; i; + 4 5i. (c) Write the following in the form a + bi. (i) i ; i ; i 4 ; i 7 ; i 98 ; i 6 ; i 4. (ii) ( i)+( 7+i); ( + 5i)+(8i 7); ( 7 + ) ( 4 i ) i.

55 .7. COMPLEX NUMBERS 5 (iii) ( i) ( 7+i); ( + 5i) (8 ( 7i); 7 + ) 4 i ( ) i. (iv) 5( 4i); 9 6; 7 ; 5i( 4i); ( i)( 7 +i); ( + 5i)(8 ( 7i); 7 + )( 4 i ) i. ( 5 (v) ( 4i) ; 5i ( 4i) ; ( i) ( 7+i) ; ( + 5i) (8 7i) ; 7 + ) 4 i ( ). i.7. Homework Exercises Write the real and imaginary parts of. 9 +i; 5+4i; 8 9i; 9 i.. 7; i; 4i. What is the complex conjugate of each of the following?. ; i; 9+i; Write the following in the form a + bi 4 8i; i. 4. i ; i ; i ; i 7 ; i ; i ( 9+i)+( 6i); ( 4 8i)+( 4 5+7i); 6. ( 9+i) ( 6i); ( 4 8i) ( 4 5+7i); ( ) 9 i ( ) 9 i ( i ( + 5 i ). ). 7. 4(+i); ( 4 5; ; 4i( +i); ( 9+i)( 6i); ( 8i)( 5+ 7i); )( 9 i + 5 ) i ( + i) ; 4i ( +i) ; ( 9+i) ( 6i) ; ( 8i) ( 5+7i) ; ( ) 9 i ( + 5 ). i

56 5 CHAPTER. REAL NUMBERS

57 Chapter Quadratic Functions. Solving Quadratic Equations By Factoring An equation is a mathematical statement involving an equality (). A quadratic equation in one variable, x, can be written in the form ax + bx + c 0 for real numbers a, b, c, a 0. To solve a quadratic equation is to find those numbers which, when substituted for x, satisfy the equation. In this section, we will solve quadratic equations in which the quadratic polynomial can be factored. Preliminaries on multiplication and factorization of polynomials are given in the appendix of this book. First, the Zero product law: When a b 0 then a 0or b 0. That is, if the product of two numbers is zero, then one or the other has to be zero. Notice that, this is true only for zero. For instance, if a b 6,thenitdoesnotmeanthata 6orb 6. Indeed, 6. Thus,zeroalonehasthisproperty. Using the Zero product law, we proceed. Examples : x(x ) 0 x 0 or x 0 (Zero product law.) x 0 or x (Solve the two linear equations.) 5

58 54 CHAPTER. QUADRATIC FUNCTIONS x(x ) 6 (We need a zero on one side of the equation.) x x 6 (Subtract 6 to get a zero.) x x 6 0 (Factor the left.) (x )(x +) 0 x 0 or x +0 (Zero product law) x or x (Solve the two linear equations.) x(x ) (x ) (We need a zero.) x(x ) + (x ) 0 (Adding (x ) to get a zero.) 6x x +x 0 (Simplify the left.) 6x x 0 (Now factor the left.) (x +)(x ) 0 x +0 or x 0 (Zero product law) x or x x or x 4x 5x ( x) (x ) (First cross-multiply.) 4x (x ) 5x ( x) (Get a zero on the right.) 4x (x ) 5x ( x) 0 (Simplify the left.) x 4 4x 5x +5x 0 x 4 +x 5x 0 (GCF x.) x (x +x 5) 0 (Factor the left.) x (4x )(x +5) 0 x 0, 4x 0, or x +50

59 .. SOLVING QUADRATIC EQUATIONS BY FACTORING 55 x 0, or4x, or x 5 x 0, orx 4, or x 5 x 0, orx 4, or x 8x 4 x x 4 x 0x +5 0 Using ac-method, with 8 5 0; 4x (x ) 5(x ) 0 (x )(4x 5) 0 The left hand side is factored; x 0 or4x 50 The Zero property; x or4x 5 x x ± 6 x ± x x 5 0 ( x) x 5 0 or x or x ± 4 5 or x ± Notice the two options ±; Rationalize the denominator. ( x 5)( x +) 0 Factoring the left hand side; x 50 or x +0 The Zero property; x 5 or x x 5 orx 4 Squaring both sides. Check which of these are solutions; x 5is a solution to this equation, but x 4is not a solution. Classroom Exercises : Solve the following equations. (a) x +x 0 (b)x +x 0 (c)6x(x +)x 4 (d) x x x (e) x 4 5x (x ) (4x ) (g) x 4 +5x (h)x 9 x 5 (f) x 4 x 6

60 56 CHAPTER. QUADRATIC FUNCTIONS.. Homework Exercises Solve the following equations: () x +7x 0 x +7x x(x +5)4(x ) () x(x +)5(x ) 6x (x +) (x ) x 6 x5 (x ) (x ) () x 4 x 8 6x 4 +x 5 x 7 x 8. Completing the square and the quadratic formula A polynomial of degree is called,a quadratic polynomial. A quadratic equation is an equation which can be written in the form of P 0foraquadraticpolynomialP.Inourcourse we will be concerned with quadratic equations with real coefficients in one variable. Any quadratic equation with real coefficients in one variable, x, can be written in the form ax + bx + c 0 fora, b, c R,a 0. Examples of quadratic equations with real coefficients are x +x +40, x +4x 50, 4 x x +70, x 5 x + 0, x +4x 0, x +0,. We derive the complete squares formulae: (x + h) (x + h)(x + h) x + hx + hx + h x +hx + h. Likewise, (x h) (x h)(x h) x hx hx + h x hx + h. Every quadratic equation can be written in terms of a complete square as follows. a(x h) t for a, h, t R,a 0. The process of converting any quadratic polynomial in one variable to a polynomial in the form of a(x h) t complete square is called completing the square.

61 .. COMPLETING THE SQUARE AND THE QUADRATIC FORMULA 57 Example : x +4x +4(x +). (this is just the complete square formula). Example : x +4x x +4x +4 }{{} 4 (x +) 4. Example : x +4x +7x +4x +4 }{{} 4+7 (x +) +. Example 4: x 8x x 8x +( 4) }{{} (4) (x 4) 6. By now the reader must have recognized the pattern. Example 5: ( ) b ( ) b x + bx x + bx + }{{} ( x + ) b ( ) b Now consider the general (that is of the form ax + )quadraticpolynomial: Example 6: x +5x (x + 5 ) x x + 5 ( ) 5 ( ) 5 x }{{} ( ( x + 5 ) ) ( since ) ; 4

62 58 CHAPTER. QUADRATIC FUNCTIONS ( x + 5 ) ( x + 5 ) (distribute multiplication by ) In general, we see the following: Example 7: ax + bx a (x + ba ) x a x + b ( ) b ( ) b ( a x + since b a a a b a b ) ; a }{{} ( ( a x + b ) ) b a 4a ( a x + b ) a b a 4a (distribute the multiplication by a) ( a x + b ) b a 4a. Example 8: ax + bx + c a (x + ba ) x + c a x + b ( ) b ( ) b a x + + c a a }{{} ( ( a x + b ) ) b a 4a + c ( a x + b ) a b a 4a + c ( a x + b ) b a 4a + c ( a x + b ) ( ) b a 4a c }{{} ( a x + b ) b 4ac a 4a

63 .. COMPLETING THE SQUARE AND THE QUADRATIC FORMULA 59 Classroom Exercises : Complete the following as squares: (a) x 6x (b) x +8x (c) x +7x (d) x +9x (e) x +6x +4 (f)x 4x (g) x 7x + (h)x +7x 4 Once we know to complete a quadratic polynomial to a square, we cansolveaquadratic equation. To solve a quadartic equation is to find the value of x which satisfies the equation. Let us consider some of the examples considered above. Example : x +4x +40 (x +) 0 (completing of the square) (x +) 00 (taking square-roots of both sides) x Example : Here we get complex solutions. Example 6: x +4x +70 (subtract from both sides). (x +) +0 (completing of the square) (x +) (subtract from both sides) (x +)± ± i (taking square-roots of both sides) x ± i (subtract from both sides). x +5x 0 ( x + 5 ) 5 0 (completing the square) 4 8 ( x + 5 ) 5 (add 5 to both sides) ( x + 5 ) (divide both sides by ) ( x + 5 ) ( x + 5 ) 5 ± 4 6