QUADRATIC FUNCTIONS AND COMPLEX NUMBERS

Transcription

1 CHAPTER 86 5 CHAPTER TABLE F CNTENTS 5- Real Roots of a Quadratic Equation 5-2 The Quadratic Formula 5-3 The Discriminant 5-4 The Comple Numbers 5-5 perations with Comple Numbers 5-6 Comple Roots of a Quadratic Equation 5-7 Sum and Product of the Roots of a Quadratic Equation 5-8 Solving Higher Degree Polnomial Equations 5-9 Solutions of Sstems of Equations and Inequalities Chapter Summar Vocabular Review Eercises Cumulative Review QUADRATIC FUNCTINS AND CMPLEX NUMBERS Records of earl Bablonian and Chinese mathematics provide methods for the solution of a quadratic equation.these methods are usuall presented in terms of directions for the solution of a particular problem, often one derived from the relationship between the perimeter and the area of a rectangle. For eample, a list of steps needed to find the length and width of a rectangle would be given in terms of the perimeter and the area of a rectangle. The Arab mathematician Mohammed ibn Musa al-khwarizmi gave similar steps for the solution of the relationship a square and ten roots of the same amount to thirt-nine dirhems. In algebraic terms, is the root, 2 is the square, and the equation can be written as His solution used these steps:. Take half of the number of roots : 2 of Multipl this number b itself: Add this to 39: Now take the root of this number:! Subtract from this number one-half the number of roots: of This is the number which we sought. In this chapter, we will derive a formula for the solution of an quadratic equation. The derivation of this formula uses steps ver similar to those used b al-khwarizmi.

2 Real Roots of a Quadratic Equation REAL RTS F A QUADRATIC EQUATIN The real roots of a polnomial function are the -coordinates of the points at which the graph of the function intersects the -ais. The graph of the polnomial function is shown at the right. The graph intersects the -ais between 22 and 2 and 2 2 between and 2.The roots of appear to be about 2.4 and.4. Can we find the eact values of these roots? Although we cannot factor over the set of integers, we can find the value of 2 and take the square roots of each side of the equation !2 5 The roots of the function are the irrational numbers 2!2 and!2. The graph of the polnomial function is shown at the right. The graph shows that the function has two real roots, one between 23 and 22 and the other between 0 and. Can we find the eact values of the roots of the function ? We know that the equation cannot be solved b factoring the right member over the set of integers. However, we can follow a process similar to that used to solve We need to write the right member as a trinomial that is the square of a binomial. This process is called completing the square. 2 2 Recall that ( h) h h 2. To write a trinomial that is a perfect square using the terms 2 2, we need to let 2h 5 2. Therefore, h 5 and h 2 5. The perfect square trinomial is 2 2, which is the square of ( ). () Add to both sides of the equation to isolate the terms in : (2) Add the number needed to complete the square to both sides of the equation: 2 5 ( ) 2 A 2 coefficient of B 2 A 2 (2) B (3) Write the square root of both sides 6!2 5 of the equation: (4) Solve for : 2 6!2 5 The roots of = are 2 2!2 and 2!2.

3 88 Quadratic Functions and Comple Numbers We can use a calculator to approimate these roots to the nearest hundredth. ENTER: (-) 2nd ENTER: (-) 2 ENTER 2 ENTER 2nd DISPLAY: - - ( DISPLAY: - + ( To the nearest hundredth, the roots are 22.4 and 0.4. ne root is between 23 and 22 and the other is between 0 and, as shown on the graph. The process of completing the square can be illustrated geometricall. Let us draw the quadratic epression 2 2 as a square with sides of length and a rectangle with dimensions 3 2. Cut the rectangle in half and attach the two parts to the sides of the square. The result is a larger square with a missing corner. The factor A 2 coefficient of B 2 or 2 is the area of this missing corner. This is the origin of the epression completing the square ( ) 2 2 Procedure To solve a quadratic equation of the form a 2 b 2 c 5 0 where a 5 b completing the square:. Isolate the terms in on one side of the equation. A 2 b B 2 2. Add the square of one-half the coefficient of or to both sides of the equation. 3. Write the square root of both sides of the resulting equation and solve for. Note: The procedure outlined above onl works when a 5. If the coefficient a is not equal to, divide each term b a, and then proceed with completing the square using the procedure outlined above.

4 Real Roots of a Quadratic Equation 89 Not ever second-degree polnomial function has real roots. Note that the graph of does not intersect the -ais and therefore has no real roots. Can we find roots b completing the square? A 2 coefficient of B 2 5 A 2 (26) B 2 5 (23) Use to complete the square Isolate the terms in Add the square of one-half the 2 5 ( 2 3) 2 coefficient of to complete square. 6! Take the square root of both sides. 3 6!2 5 Solve for. Since there is no real number that is the square root of 2, this function has no real zeros. We will stud a set of numbers that includes the square roots of negative numbers later in this chapter. EXAMPLE Find the eact values of the zeros of the function f() Solution How to Proceed () Let f() 5 0: (2) Isolate the terms in : (3) Multipl b 2: (4) Complete the square b adding A 2 coefficient of B 2 A 2 (4) B to both sides of the equation: (5) Take the square root of each side of the equation: (6) Solve for : ( 2) 2 6! !6 5 Answer The zeros are (22!6) and (22 2!6).

5 90 Quadratic Functions and Comple Numbers EXAMPLE 2 Find the roots of the function Solution How to Proceed () Let 5 0: (2) Isolate the terms in : (3) Divide both sides of the equation b the coefficient of 2 : (4) Complete the square b adding A 2 coefficient of B 2 5 A 2? 2 2 B to both sides of the equation: (5) Take the square root of both sides of the equation: (6) Solve for : # A 2 4 B A 2 4 B A 2 4 B 2 5 The roots of the function are: Answer 2 2 and and Note that since the roots are rational, the equation could have been solved b factoring over the set of integers (2 )( 2 ) Graphs of Quadratic Functions and Completing the Square A quadratic function 5 a( 2 h) 2 k can be graphed in terms of the basic quadratic function 5 2. In particular, the graph of 5 a( 2 h) 2 k is the graph of 5 2 shifted h units horizontall and k units verticall. The coefficient a determines its orientation: when a. 0, the parabola opens upward; when

6 Real Roots of a Quadratic Equation 9 a, 0, the parabola opens downward.the coefficient a also determines how narrow or wide the graph is compared to 5 2 : when a., the graph is narrower; when a,, the graph is wider. We can use the process of completing the square to help us graph quadratic functions of the form 5 a 2 b c, as the following eamples demonstrate: EXAMPLE 3 Solution EXAMPLE 4 Graph f() The square of one-half the coefficient of is or 4. A 2? 4 B 2 Add and subtract 4 to keep the function equivalent to the original and complete the square. f() ( 2 4 4) 5 5 ( 2) 2 5 The graph of f() is the graph of 5 2 shifted 2 units to the left and 5 units up, as shown on the right. Note that this function has no real roots. Graph f() f() 2 Solution 2 f() How to Proceed () Factor 2 from the first two terms: (2) Add and subtract A 2 coefficient of B 2 5 A 2? 2 B 2 5 to keep the function equivalent to the original: (3) Complete the square: f() ( 2 2) 5 2( ) 5 2( 2 2 ) ( 2 2 ) 2 5 2( ) 2 2 The graph of f() is the graph of 5 2 stretched verticall b a factor of 2, and shifted unit to the left and unit down as shown on the left. Note that this function has two real roots, one between 22 and 2 and the other between 2 and 0.

7 92 Quadratic Functions and Comple Numbers Eercises Writing About Mathematics. To solve the equation given in step 2 of Eample 2, Sarah multiplied each side of the equation b 8 and then added to each side to complete the square. Show that is the square of a binomial and will lead to the correct solution of Phillip said that the equation can be solved b adding 8 to both sides of the equation. Do ou agree with Phillip? Eplain wh or wh not. Developing Skills In 3 8, complete the square of the quadratic epression In 9 4: a. Sketch the graph of each function. b. From the graph, estimate the roots of the function to the nearest tenth. c. Find the eact irrational roots in simplest radical form. 9. f() f() f() f() f() f() In 9 26, solve each quadratic equation b completing the square. Epress the answer in simplest radical form a. Complete the square to find the roots of the equation b. Write, to the nearest tenth, a rational approimation for the roots. In 28 33, without graphing the parabola, describe the translation, reflection, and/or scaling that must be applied to 5 2 to obtain the graph of each given function. 28. f() f() f() f() f() f() Determine the coordinates of the verte and the equation of the ais of smmetr of f() b writing the equation in the form f() 5 ( 2 h) 2 k. Justif our answer.

8 The Quadratic Formula 93 Appling Skills 35. The length of a rectangle is 4 feet more than twice the width. The area of the rectangle is 38 square feet. a. Find the dimensions of the rectangle in simplest radical form. b. Show that the product of the length and width is equal to the area. c. Write, to the nearest tenth, rational approimations for the length and width. 36. ne base of a trapezoid is 8 feet longer than the other base and the height of the trapezoid is equal to the length of the shorter base. The area of the trapezoid is 20 square feet. a. Find the lengths of the bases and of the height of the trapezoid in simplest radical form. b. Show that the area of the trapezoid is equal to one-half the height times the sum of the lengths of the bases. c. Write, to the nearest tenth, rational approimations for the lengths of the bases and for the height of the trapezoid. 37. Steve and Alice realized that their ages are consecutive odd integers. The product of their ages is 95. Steve is ounger than Alice. Determine their ages b completing the square. 5-2 THE QUADRATIC FRMULA We have solved equations b using the same steps to complete the square needed to find the roots or zeros of a function of the form 5 a 2 b c, that is, to solve the quadratic equation 0 5 a 2 b c. We will appl those steps to the general equation to find a formula that can be used to solve an quadratic equation. How to Proceed () Let 5 0: 0 5 a 2 + b + c (2) Isolate the terms in : 2c 5 a 2 b (3) Divide both sides of the equation b the coefficient of 2 : (4) Complete the square b adding A 2 coefficient of B 2 5 A 2? b a B 2 5 b2 4a 2 to both sides of the equation: (5) Take the square root of each side of the equation: (6) Solve for : This result is called the quadratic formula. b 2 4a 2c 2 a 5 2 b a b2 4a 2 b 2 4a 24ac 2 4a 5 2 b 2 a b2 4a 2 b 2 2 4ac 4a 2 5 A b 2a B 2 6# b2 2 4ac 4a 2 2b 2a 6 "b2 2 4ac 2a 2b 6 "b 2 2 4ac 2a When a 0, the roots of a 2 2b 6 "b b c 5 0 are ac 2a. 2c a 5 2 b a b 2a

9 94 Quadratic Functions and Comple Numbers EXAMPLE Use the quadratic formula to find the roots of Solution Answer How to Proceed () Write the equation in standard form: (2) Determine the values of a, b, and c: (3) Substitute the values of a, b, and c in the quadratic formula: (4) Perform the computation: (5) Write the radical in simplest form: (6) Divide numerator and denominator b 2: 2!6 2 and 2 2! a 5 2, b 524, c 52 2b 6 "b ac 2a 2(24) 6 "(24) (2)(2) 2(2) 4 6! ! !4! ! !6 EXAMPLE 2 Use the quadratic formula to show that the equation has no real roots. Solution For the equation , a 5, b 5, and c b 6 "b ac 2a " 2 2 4()(2) 2 6! !27 2() There is no real number that is the square root of a negative number. Therefore,!27 is not a real number and the equation has no real roots. EXAMPLE 3 ne leg of a right triangle is centimeter shorter than the other leg and the hpotenuse is 2 centimeters longer than the longer leg. What are lengths of the sides of the triangle?

10 The Quadratic Formula 95 Solution Let 5 the length of the longer leg 2 5 the length of the shorter leg 2 5 the length of the hpotenuse Use the Pthagorean Theorem to write an equation. 2 ( 2 ) 2 5 ( 2) Use the quadratic formula to solve this equation: a 5, b 526, c b 6 "b ac 2a 2(26) 6 "(26) ()(23) 2() 6 6! ! !6! ! !3 The roots are 3 2!3 and 3 2 2!3. Reject the negative root, 3 2 2!3. The length of the longer leg is 3 2!3. The length of the shorter leg is 3 2!3 2 or 2 2!3. The length of the hpotenuse is 3 2!3 2 or 5 2!3. Check A3 2!3B 2 A2 2!3B 2 5? A5 2!3B 2 A9 2!3 4? 3B A4 8!3 4? 3B 5? 25 20!3 4? ! !3 Eercises Writing About Mathematics 2!6 2 2!6. Noah said that the solutions of Eample, 2 and 2, could have been written as 2!6 2 2!6 2 5!6 and 2 5 2!6. Do ou agree with Noah? Eplain wh or wh not. 2. Rita said that when a, b, and c are real numbers, the roots of a 2 b + c 5 0 are real numbers onl when b 2 $ 4ac. Do ou agree with Rita? Eplain wh or wh not.

11 96 Quadratic Functions and Comple Numbers Developing Skills In 3 4, use the quadratic formula to find the roots of each equation. Irrational roots should be written in simplest radical form a. Sketch the graph of b. From the graph, estimate the roots of the function to the nearest tenth. c. Use the quadratic formula to find the eact values of the roots of the function. d. Epress the roots of the function to the nearest tenth and compare these values to our estimate from the graph. Appling Skills In 9 25, epress each answer in simplest radical form. Check each answer. 9. The larger of two numbers is 5 more than twice the smaller. The square of the smaller is equal to the larger. Find the numbers. 20. The length of a rectangle is 2 feet more than the width. The area of the rectangle is 2 square feet. What are the dimensions of the rectangle? 2. The length of a rectangle is 4 centimeters more than the width. The measure of a diagonal is 0 centimeters. Find the dimensions of the rectangle. 22. The length of the base of a triangle is 6 feet more than the length of the altitude to the base. The area of the triangle is 8 square feet. Find the length of the base and of the altitude to the base. 23. The lengths of the bases of a trapezoid are 0 and 3 2 and the length of the altitude is 2. If the area of the trapezoid is 40, find the lengths of the bases and of the altitude. 24. The altitude, CD, to the hpotenuse, AB, of right triangle ABC separates the hpotenuse into two segments, AD and DB. If AD 5 DB 4 and CD 5 2 centimeters, find DB, AD, and AB. Recall that the length of the altitude to the hpotenuse of a right triangle is the mean proportional between the lengths of the segments into which the hpotenuse is separated, that is, AD CD 5 DB CD. A C DB 4 2 cm D B

12 The Quadratic Formula A parabola is smmetric under a line reflection. Each real root of the quadratic function 5 a 2 b + c is the image of the other under a reflection in the ais of smmetr of the parabola. a. What are the coordinates of the points at which the parabola whose equation is 5 a 2 b + c intersects the -ais? b. What are the coordinates of the midpoint of the points whose coordinates were found in part a? c. What is the equation of the ais of smmetr of the parabola 5 a 2 b + c? d. The turning point of a parabola is on the ais of smmetr. What is the -coordinate of the turning point of the parabola 5 a 2 b + c? 26. Gravit on the moon is about one-sith of gravit on Earth. An astronaut standing on a tower 20 feet above the moon s surface throws a ball upward with a velocit of 30 feet per second. The height of the ball at an time t (in seconds) is h(t) t 2 30t 20. To the nearest tenth of a second, how long will it take for the ball to hit the ground? 27. Based on data from a local college, a statistician determined that the average student s grade point average (GPA) at this college is a function of the number of hours, h, he or she studies each week. The grade point average can be estimated b the function G(h) h h.2 for 0 # h # 20. a. To the nearest tenth, what is the average GPA of a student who does no studing? b. To the nearest tenth, what is the average GPA of a student who studies 2 hours per week? c. To the nearest tenth, how man hours of studing are required for the average student to achieve a 3.2 GPA? 28. Follow the steps given in the chapter opener to find a root of the equation 2 b c a a 5 0. Compare our solution with the quadratic formula. Hands-n Activit: Alternate Derivation of the Quadratic Formula As ou ma have noticed, the derivation of the quadratic formula on page 93 uses fractions throughout (in particular, starting with step 3). We can use an alternate derivation of the quadratic equation that uses fractions onl in the last step. This alternate derivation uses the following perfect square trinomial: 4a 2 2 4ab b 2 5 4a 2 2 2ab 2ab + b 2 5 2a(2a + b) b(2a + b) 5 (2a + b)(2a + b) 5 (2a + b) 2

13 98 Quadratic Functions and Comple Numbers To write a 2 b, the terms in of the general quadratic equation, as this perfect square trinomial, we must multipl b 4a and add b 2. The steps for the derivation of the quadratic formula using this perfect square trinomial are shown below. How to Proceed () Isolate the terms in : a 2 b + c 5 0 a 2 b 52c (2) Multipl both sides of the equation b 4a: 4a 2 2 4ab 524ac (3) Add b 2 to both sides of the equation to complete 4a 2 2 4ab b 2 5 b 2 2 4ac the square: (2a b) 2 5 b 2 2 4ac (4) Now solve for. Does this procedure lead to the quadratic formula? 5-3 THE DISCRIMINANT When a quadratic equation a 2 b c 5 0 has rational numbers as the values of a, b, and c, the value of b 2 2 4ac, or the discriminant, determines the nature of the roots. Consider the following equations. CASE The discriminant b 2 2 4ac is a positive perfect square where a 5 2, b 5 3, and c b 6 "b2 2 4ac 2a "32 2 4(2)(22) 2(2) " The roots are and The roots are unequal rational numbers, and the graph of the corresponding quadratic function has two -intercepts. CASE 2 The discriminant b 2 2 4ac is a positive number that is not a perfect square where a 5 3, b 5, and c b 6 "b 2 2 4ac 2a 2 6 " 2 2 4(3)(2) 2(3) 2 6! " !3 2 2!3 The roots are 6 and !

14 The Discriminant 99 The roots are unequal irrational numbers, and the graph of the corresponding quadratic function has two -intercepts. CASE 3 The discriminant b 2 2 4ac is where a 5 9, b 526, and c b 6 "b 2 2 4ac 2a 2(26) 6 "(26) 2 2 4(9)() 2(9) 6 6 " " The roots are and The roots are equal rational numbers, and the graph of the corresponding quadratic function has one -intercept. The root is a double root CASE 4 The discriminant b 2 2 4ac is a negative number where a 5 2, b 5 3, and c b 6 "b 2 2 4ac 2a 23 6 " (2)(4) 2(2) 23 6! This equation has no real roots because there is no real number equal to!223, and the graph of the corresponding quadratic function has no -intercepts ! In each case, when a, b, and c are rational numbers, the value of b 2 2 4ac determined the nature of the roots and the -intercepts. From the four cases given above, we can make the following observations. Let a 2 b c 5 0 be a quadratic equation and a, b, and c be rational numbers with a 0: The number When the discriminant The roots of the of -intercepts b 2 2 4ac is: equation are: of the function is:. 0 and a perfect square real, rational, and unequal 2. 0 and not a perfect square real, irrational, and unequal real, rational, and equal, 0 not real numbers 0

15 200 Quadratic Functions and Comple Numbers EXAMPLE Without solving the equation, find the nature of the roots of Solution Write the equation in standard form: Evaluate the discriminant of the equation when a 5 3, b 5 2, c 52. b 2 2 4ac (3)(2) Since a, b, and c are rational and the discriminant is greater than 0 and a perfect square, the roots are real, rational, and unequal. EXAMPLE 2 Show that it is impossible to draw a rectangle whose area is 20 square feet and whose perimeter is 0 feet. Solution Let l and w be the length and width of the rectangle. How to Proceed () Use the formula for perimeter: 2l 2w 5 0 (2) Solve for l in terms of w: l w 5 5 l w (3) Use the formula for area: (4) Substitute 5 2 w for l. Write the resulting quadratic equation in standard form: lw 5 20 (5 2 w)(w) w 2 w w 2 5w w 2 2 5w (5) Test the discriminant: b 2 2 4ac 5 (25) 2 2 4()(20) Since the discriminant is negative, the quadratic equation has no real roots. There are no real numbers that can be the dimensions of this rectangle. EXAMPLE 3 A lamp manufacturer earns a weekl profit of P dollars according to the function P() where is the number of lamps sold.when P() 5 0, the roots of the equation represent the level of sales for which the manufacturer will break even.

16 The Discriminant 20 a. Interpret the meaning of the discriminant value of 2,296. b. Use the discriminant to determine the number of break even points. c. Use the discriminant to determine if there is a level of sales for which the weekl profit would be $3,000. Solution a. Since the discriminant is greater than 0 and not a perfect square, the equation has two roots or break even points that are irrational and unequal. Answer b. 2 (see part a) Answer c. We want to know if the equation 3, has at least one real root. Find the discriminant of this equation. The equation in standard form is ,70. b 2 2 ac (20.3)(23,70) 52,304 The discriminant is negative, so there are no real roots. The compan cannot achieve a weekl profit of $3,000. Answer Eercises Writing About Mathematics. a. What is the discriminant of the equation 2 A!5B 2 5 0? b. Find the roots of the equation 2 A!5B c. Do the rules for the relationship between the discriminant and the roots of the equation given in this chapter appl to this equation? Eplain wh or wh not. 2. Christina said that when a, b, and c are rational numbers and a and c have opposite signs, the quadratic equation a 2 b + c 5 0 must have real roots. Do ou agree with Christina? Eplain wh or wh not. Developing Skills In 3 8, each graph represents a quadratic function. Determine if the discriminant of the quadratic function is greater than 0, equal to 0, or less than

17 202 Quadratic Functions and Comple Numbers In 9 4: a. For each given value of the discriminant of a quadratic equation with rational coefficients, determine if the roots of the quadratic equation are () rational and unequal, (2) rational and equal, (3) irrational and unequal, or (4) not real numbers. b. Use our answer to part a to determine the number of -intercepts of the graph of the corresponding quadratic function In 5 23: a. Find the value of the discriminant and determine if the roots of the quadratic equation are () rational and unequal, (2) rational and equal, (3) irrational and unequal, or (4) not real numbers. b. Use an method to find the real roots of the equation if the eist When b 2 2 4ac 5 0, is a 2 b c a perfect square trinomial or a constant times a perfect square trinomial? Eplain wh or wh not. 25. Find a value of c such that the roots of 2 2 c 5 0 are: a. equal and rational. b. unequal and rational. c. unequal and irrational. d. not real numbers. 26. Find a value of b such that the roots of 2 b are: a. equal and rational. b. unequal and rational. c. unequal and irrational. d. not real numbers. Appling Skills 27. Lauren wants to fence off a rectangular flower bed with a perimeter of 30 ards and a diagonal length of 8 ards. Use the discriminant to determine if her fence can be constructed. If possible, determine the dimensions of the rectangle.

18 28. An open bo is to be constructed as shown in the figure on the right. Is it possible to construct a bo with a volume of 25 cubic feet? Use the discriminant to eplain our answer. The Comple Numbers The height, in feet, to which of a golf ball rises when shot upward from ground level is described b the function h(t) 526t 2 48t where t is the time elapsed in seconds. Use the discriminant to determine if the golf ball will reach a height of 32 feet. 30. The profit function for a compan that manufactures cameras is P() ,000. Under present conditions, can the compan achieve a profit of $20,000? Use the discriminant to eplain our answer. 5 4 ft 5-4 THE CMPLEX NUMBERS The Set of Imaginar Numbers We have seen that when we solve quadratic equations with rational coefficients, some equations have real, rational roots, some have real, irrational roots, and some do not have real roots. For eample: !9 5 6!8 5 6! !4!2 5 6!9!2 5 62!2 5 63!2 The equations and have real roots, but the equation does not have real roots. A number is an abstract idea to which we assign a meaning and a smbol. We can form a new set of numbers that includes!2, to which we will assign the smbol i. We call i the unit of the set of imaginar numbers. The numbers 3!2 and 23!2 that are the roots of the equation can be written as 3i and 23i. DEFINITIN A number of the form a!2 5 ai is a pure imaginar number where a is a non-zero real number. The idea of the square root of 2 as a number was not accepted b man earl mathematicians. Rafael Bombelli ( ) introduced the terms plus of minus and minus of minus to designate these numbers. For instance, he

19 204 Quadratic Functions and Comple Numbers designated 2 3i as 2 plus of minus 3 and 2 2 3i as 2 minus of minus 3.The term imaginar was first used in 637 b René Descartes, who used it in an almost derogator sense, impling that somehow imaginar numbers had no significance in mathematics. However, the algebra of imaginar numbers was later developed and became an invaluable tool in real-world applications such as alternating current. Some elements of the set of pure imaginar numbers are:!225 5!25!2 5 5i 2!26 52!4!2 522i!20 5!0!2 5!2!0 5 i!0 2!232 52!6!2!2 524!2!2 524i!2 Note the order in which the factors of i!0 and 24i!2 are written. The factor that is a radical is written last to avoid confusion with!0i and 24!2i. EXAMPLE Simplif: a.!27 b.!22 Solution Write each imaginar number in terms of i and simplif. a.!27 5!7!2 5 i!7 b.!22 5!4!3!2 5 2i!3 Powers of i The unit element of the set of pure imaginar numbers is i 5!2. Since!b is one of the two equal factors of b, then i 5!2 is one of the two equal factors of 2 and i i 52 or i The bo at the right shows the first four powers of i. In general, for an integer n: i 5!2 i 2 52 i 3 5 i 2 i 52i 52i i 4 5 i 2 i 2 52(2) 5 i 4n 5 (i 4 ) n 5 n 5 i 4n 5 (i 4 ) n (i ) 5 n (i) 5 (i) 5 i i 4n2 5 (i 4 ) n (i 2 ) 5 n (2) 5 (2) 5 2 i 4n3 5 (i 4 ) n (i 3 ) 5 n (2i) 5 (2i) 5 2i For an positive integer k, i k is equal to i, 2, 2i, or. We can use this relationship to evaluate an power of i.

20 The Comple Numbers 205 EXAMPLE 2 Evaluate: Answers a. i 4 5 i 4(3)2 5 (i 4 ) 3 (i 2 ) 5 () 3 (2) 5 (2) 52 b. i 7 5 i 7 5 i 43 5 i 4? i 3 5 ()(2i) 52i Arithmetic of Imaginar Numbers The rule for multipling radicals,!a?!b 5!ab, is true if and onl if a and b are non-negative numbers. When a, 0 or b, 0,!a?!b 2!ab. For eample, it is incorrect to write:!24 3!29 5!36 5 6, but it is correct to write:!24 3!29 5!4!2 3!9!2 5 2i 3 3i 5 6i 2 5 6(2) 526 The distributive propert of multiplication over addition is true for imaginar numbers. 5 5i!24!29 5!4!2!9!2 5 2i 3i 5 (2 3)i!28!250 5!4!2!2!25!2!2 5 2i!2 5i!2 5 (2 5)Ai!2B 5 7i!2 EXAMPLE 3 Find the sum of!225!249. Solution Write each imaginar number in terms of i and add similar terms.!225!249 5!25!2!49!2 5 5i 7i 5 2i Answer The Set of Comple Numbers The set of pure imaginar numbers is not closed under multiplication because the powers of i include both real numbers and imaginar numbers. For eample, i i 5 2. We want to define a set of numbers that includes both real numbers and pure imaginar numbers and that satisfies all the properties of the real numbers. This new set of numbers is called the set of comple numbers and is defined as the set of numbers of the form a + bi where a and b are real numbers. When a 5 0, a bi 5 0 bi 5 bi, a pure imaginar number. When b 5 0, a bi 5 a 0i 5 a, a real number. Therefore, both the set of pure imaginar numbers and the set of real numbers are contained in the set of comple numbers.a comple number that is not a real number is called an imaginar number.

21 206 Quadratic Functions and Comple Numbers DEFINITIN A comple number is a number of the form a bi where a and b are real numbers and i 5!2. Eamples of comple numbers are: 2, 24, 3,!2, p, 4i, 25i, 4 i, i!5, pi, 2 2 3i, 27 i, 8 3i, 0 2 i, 5 0i Comple Numbers and the Graphing Calculator The TI famil of graphing calculators can use comple numbers. To change our calculator to work with comple numbers, press MDE, scroll down to the row that lists REAL, select a bi, and then press ENTER. Your calculator will now evaluate comple numbers. For eample, with our calculator in a bi mode, write 6!29 as a comple number in terms of i. ENTER: 6 2nd 29 ) ENTER DISPLAY: 6+ ( - 9) 6+3i EXAMPLE 4 Evaluate each epression using a calculator: a.!225 2!26 b. 2!29 5!24 Calculator Solution a. Press 2nd ) ENTER. ) 2nd ( - 25)- ( - 6) i b. Press 2 2nd 29 ) 5 2nd 24 ) ENTER. 2+ ( - 9)+5 ( - 4) 2+3i Answers a. i b. 2 3i

22 The Comple Numbers 207 What about powers of i? A calculator is not an efficient tool for evaluating an epression such as bi n for large values of n. Large comple powers introduce rounding errors in the answer as shown in the calculator screen on the right. i 7 < - 3e - 3-i Graphing Comple Numbers We are familiar with the one-to-one correspondence between the set of real numbers and the points on the number line. The comple numbers can be put in a one-to-one correspondence with the points of a plane. This correspondence makes it possible for a phsical quantit, such as an alternating current that has both amplitude and direction, to be represented b a comple number. n graph paper, draw a horizontal real number line. Through 0 on the real number line, draw a vertical line. This vertical line is the pure imaginar number line. n this line, mark points equidistant from one another such that the distance between them is equal to the unit distance on the real number line. Label the points above 0 as i, 2i, 3i, 4i, and so on and the numbers below 0 as 2i, 22i, 23i, 24i, and so on. To find the point that corresponds to the comple number 4 2 3i, move to 4 on the real number line and then, parallel to the imaginar number line, move down 3 units to a point on the same horizontal line as 3i i 3i 2i i i 2i 3i 4i Hands-n Activit n the comple plane, locate the points that correspond to 4 2i and 2 2 5i. Draw a line segment from 0 to each of these points. Now draw a parallelogram with these two line segments as adjacent sides. What is the comple number that corresponds to the verte of the parallelogram opposite 0? How is this comple number related to 4 2i and 2 2 5i? Repeat the procedure given above for each of the following pairs of comple numbers.. 2 4i,3 i i, 25 2i i,22 2i i,32 5i 5. 4i,2 4i i, i i,02 4i i, 24 3i 9. 4, 2i

23 208 Quadratic Functions and Comple Numbers Eercises Writing About Mathematics. Pete said that!22 3!28 5! Do ou agree with Pete? Eplain wh or wh not. 2. Ethan said that the square of an pure imaginar number is a negative real number. Do ou agree with Ethan? Justif our answer. Developing Skills In 3 8, write each number in terms of i. 3.!24 4.!28 5.! ! !22 8.!28 9.! ! ! !25 4.! !25 6.! ! !29 In 9 34, write each sum or difference in terms of i. 9.!200!28 20.!225 2!24 2.!244!2 22.!249 2!26 23.!22!227 2! !25! !4!24!236 2!36 26.!50!22!200! !4!232! ! ! !20 2 2! !26 7 2! ! ! # # # # !0.2025!20.09 In 35 43, write each number in simplest form i 2i i 2 2i i 2 5i i 5 7i i + i 3 i i 5i 8 6i 3 2i 4 4. i 2 i 2 i i + i 2 i 3 i i 57 In 44 5, locate the point that corresponds to each of the given comple numbers i i i i i i i i

24 perations with Comple Numbers 209 Appling Skills 52. Imaginar numbers are often used in electrical engineering. The impedance Z (measured in ohms) of a circuit measures the resistance of a circuit to alternating current (AC) electricit. For two AC circuits connected in series, the total impedance is the sum Z Z 2 of the individual impedances. Find the total impedance if Z 5 5i ohms and 5 8i ohms. 53. In certain circuits, the total impedance is given b the formula Z T 5 Z Z 2 Z T Z. Find Z 2 when Z 523i and Z 2 5 4i. Z 2 Z T Hands-n Activit: Multipling Comple Numbers Multiplication b i In the comple plane, the image of under a rotation of 90 about the origin is i, which is equivalent to multipling b i. Now rotate i b 90 about the origin. The image of i is 2, which models the product i i 52. This result is true for an comple number. That is, multipling an comple number a bi b i is equivalent to rotating that number b 90 about the origin. Use this graphical representation of multiplication b i to evaluate the following products:. (2 3i) i 2. 2i i 3. A i B? i 4. 25i i i 6. (23 2 2i) i Multiplication b a real number In the comple plane, the image of an number a under a dilation of k is ak, which is equivalent to multipling a b k. The image an imaginar number ai under a dilation of k is aki, which is equivalent to multipling ai b k.this result is true for an comple number.that is, multipling an comple number a bi b a real number k is equivalent to a dilation of k. Use this graphical representation of multiplication b a real number to evaluate the following products:. (2 4i) 4 2. (25 2 5i) 2 3. A 3 4 2i B? i i 2 6. (23 2 2i) 3i 5 (23 2 2i) 3 i 5-5 PERATINS WITH CMPLEX NUMBERS Like the real numbers, the comple numbers are closed, commutative, and associative under addition and multiplication, and multiplication is distributive over addition.

25 20 Quadratic Functions and Comple Numbers Adding Comple Numbers We can add comple numbers using the commutative, associative, and distributive properties that we use to add real numbers. For eample: (4 3i) (2 5i) 5 4 (3i 2) 5i Associative Propert 5 4 (2 3i) 5i Commutative Propert 5 (4 2) (3i 5i) Associative Propert 5 (4 2) (3 5)i Distributive Propert 5 6 8i Substitution Propert When we know that each of these steps leads to a correct sum, we can write: (a + bi) (c + di) 5 (a + c) (b + d)i Here are some other eamples of the addition of comple numbers: (8 2 i) (3 2 7i) 5 (8 3) (2 2 7)i 5 2 8i (3 2i) ( 2 2i) 5 (3 ) (2 2 2)i 5 4 0i 5 4 (29 i) (9 6i) 5 (29 9) ( 6)i 5 0 7i 5 7i Note that the sum of two comple numbers is alwas a comple number. That sum ma be an element of one of the subsets of the comple numbers, that is, a real number or a pure imaginar number. Since for an comple number a bi: (a bi) (0 0i) 5 (0 0i) (a bi) 5 (a bi) the real number (0 0i) or 0 is the additive identit. The additive identit of the comple numbers is the real number (0 0i) or 0. Similarl, since for an comple number a bi: (a + bi) (2a 2 bi) 5 [a (2a)] [b (2b)]i 5 0 0i 5 0 the additive inverse of an comple number (a bi) is (2a 2 bi). The additive inverse of an comple number (a bi) is (2a 2 bi). n some calculators, the ke for i is the 2nd function of the decimal point ke. When the calculator is in a bi mode, we can add comple numbers. ENTER: 4 2 2nd i 3 5 2nd i ENTER DISPLAY: 4+2i +3-5i 7-3i

26 perations with Comple Numbers 2 EXAMPLE Epress the sum in a + bi form: (23 7i) (25 2i) Solution Calculator Solution (23 7i) (25 2i) 5 (23 2 5) (7 2)i = 28 9i ENTER: ( nd i ) ( nd i ) ENTER DISPLAY: ( - 3+7i )+( - 5+2i ) - 8+9i Answer 28 9i Subtracting Comple Numbers In the set of real numbers, a 2 b 5 a (2b), that is, a 2 b is the sum of a and the additive inverse of b. We can appl the same rule to the subtraction of comple numbers. EXAMPLE 2 Subtract: Answers a. (5 3i) 2 (2 8i) 5 (5 3i) (22 2 8i) 5 (5 2 2) (3 2 8)i i b. (7 4i) 2 (2 3i) 5 (7 4i) ( 2 3i) 5 (7 ) (4 2 3)i 5 8 i 5 8 i c. (6 i) 2 (3 i) 5 (6 i) (23 2 i) 5 (6 2 3) ( 2 )i 5 3 0i 5 3 d. ( 2i) 2 ( 2 9i) 5 ( 2i) (2 9i) 5 ( 2 ) (2 9)i 5 0 i 5 i Multipling Comple Numbers We can multipl comple numbers using the same principles that we use to multipl binomials. When simplifing each of the following products, recall that i(i) 5 i 2 52.

27 22 Quadratic Functions and Comple Numbers EXAMPLE 3 Multipl: a. (5 2i)(3 4i) b. (3 2 5i)(2 i) c. (9 3i)(9 2 3i) d. (2 6i) 2 Solution a. (5 2i)(3 4i) b. (3 2 5i)(2 i) 5 5(3 4i) 2i(3 4i) 5 3(2 i) 2 5i(2 i) i 6i 8i i 0i 2 5i i 8(2) i 2 5(2) i i i Answer 5 2 7i Answer c. (9 3i)(9 2 3i) d. 5 9(9 2 3i) 3i(9 2 3i) i 27i 2 9i i 2 9(2) i 2i 36i i 36(2) 5 90 Answer i Answer In part c of Eample 3, the product is a real number. Since for an comple number a bi: (a bi)( 0i) 5 a( 0i) bi( 0i) 5 a 0i bi 0i 5 a bi the real number 0i or is the multiplicative identit. The multiplicative identit of the comple numbers is the real number 0i or. Does ever non-zero comple number have a multiplicative inverse? In the set of rational numbers, the multiplicative inverse of a is a. We can sa that in the set of comple numbers, the multiplicative inverse of a bi is a bi. To epress a bi in the form of a comple number, multipl numerator and denominator b the comple conjugate of a bi, a 2 bi. The reason for doing so is that the product of a comple number and its conjugate is a real number. For eample, ( 2i)( 2 2i) 5 ( 2 2i) 2i( 2 2i) 5 2 2i 2i 2 4i (2) 5 5 (2 6i) 2 5 (2 6i)(2 6i) 5 2(2 6i) 6i(2 6i)

28 perations with Comple Numbers 23 In general, (a bi)(a 2 bi) 5 a(a 2 bi) bi(a 2 bi) 5 a 2 2 abi abi 2 b 2 i 2 5 a 2 2 b 2 (2) 5 a 2 b 2 or (a bi)(a 2 bi) 5 a 2 b 2 The product of a comple number a + bi and its conjugate a 2 bi is a real number, a 2 b 2. Thus, if we the multipl numerator and denominator of a bi b the comple conjugate of the denominator, we will have an equivalent fraction with a denominator that is a real number. a bi? a a 2 2 bi bi 5 a 2 bi a 2 b 5 2 For eample, the comple conjugate of 2 2 4i is 2 4i and the multiplicative inverse of 2 2 4i is i i a a 2 b 2 2 We can check that 0 5 i is the multiplicative inverse of 2 2 4i b multipling the two numbers together. If 0 5 i is indeed the multiplicative inverse, then their product will be. A 0 5 i B (2 2 4i) 2 A 2 4i A 5? 0 5 i B 0 5 i B 2 A 0 B 2 A 5 i B 2 4i A 0 B 2 4i A 5 i B b a 2 b 2 i 2 2 4i i? i 4i 5 2 4i i ? i i i2 5 5? i i (2) ? 5? For an non-zero comple number a bi, its multiplicative inverse is a bi or a a 2 b 2 b 2 a 2 b i 2

29 24 Quadratic Functions and Comple Numbers EXAMPLE 4 A i B Epress the product in a + bi form: (2 3i) Solution A i B (2 3i) 5 3 (2 3i) 2 2i(2 3i) 5 4 i 2 6i i i (2) i i Calculator Solution ENTER: ( 3 2 2nd i ) ( 2 3 2nd i ) MATH ENTER ENTER DISPLAY: (/3-/2i)(2+3i ) Frac /2-5i Answer 2 2 5i Dividing Comple Numbers 2 3i We can write (2 3i) 4 ( 2i) as 2i. However, this is not a comple number of the form a bi. Can we write 2i as an equivalent fraction with a 2 3i rational denominator? In Chapter 3, we found a wa to write a fraction with an irrational denominator as an equivalent fraction with a rational denominator. We can use a similar procedure to write a fraction with a comple denominator as an equivalent fraction with a real denominator. 2 3i To write the quotient 2i with a denominator that is a real number, multipl numerator and denominator of the fraction b the comple conjugate of the denominator. The comple conjugate of 2i is 2 2i. 2 3i 2i? 2 2 2i 2 2 4i 3i 2 6i2 2i (2i) i 2 6i i i 2 6(2) 2 4(2) i i 2 3i 8 The quotient (2 3i) 4 ( 2i) 5 2i i is a comple number in a bi form.

30 perations with Comple Numbers 25 EXAMPLE 5 Solution Calculator Solution Epress the quotient in a bi form: How to Proceed () Multipl numerator and denominator b 6 to epress the denominator in terms of integers: (2) Multipl numerator and denominator of the fraction b the comple conjugate of the denominator, 3 4i: (3) Replace i 2 b its equal, 2: (4) Combine like terms: (5) Divide numerator and denominator b 25: ENTER: ( 5 0 2nd i ) ( nd i ) ENTER 5 0i i 5 0i i 5 5 0i i? i 3 2 4i 30 60i 3 2 4i? 3 4i 90 20i 80i 240i2 3 4i i 80i 240(2) i 80i i i Answer DISPLAY: (5+0i)/(/2-2/3 i ) - 6+2i Eercises Writing About Mathematics. Tim said that the binomial 2 6 can be written as 2 2 6i 2 and factored over the set of comple numbers. Do ou agree with Tim? Eplain wh or wh not. 2. Joshua said that the product of a comple number and its conjugate is alwas a real number. Do ou agree with Joshua? Eplain wh or wh not. Developing Skills In 3 7, find each sum or difference of the comple numbers in a + bi form. 3. (6 7i) ( 2i) 4. (3 2 5i) (2 i) 5. (5 2 6i) (4 2i) 6. (23 3i) 2 ( 5i) 7. (2 2 8i) 2 (2 8i) 8. (4 2i) (24 2 2i)

31 26 Quadratic Functions and Comple Numbers 9. ( 9i) 2 ( 2i) 0. (0 2 2i) 2 (2 7i). (0 3i) (0 2 3i) 2. A 2 2 i B A i B 3. A i B (2 2 i) 4. ( 0i) 2 A i B 5. A i B 2 A i B 6. A 4 2i B A 7 0 i B 7. In 8 25, write the comple conjugate of each number i i i i i i i 25. p2i In 26 37, find each product. 26. ( 5i)( 2i) 27. (3 2i)(3 3i) 28. (2 2 3i) (4 2 5i)(3 2 2i) 30. (7 i)(2 3i) 3. (22 2 i)( 2i) 32. (24 2 i)(24 i) 33. (22 2 2i)(2 2 2i) 34. (5 2 3i)(5 3i) 35. (3 2 i) 36. (4 2i) 37. (4 4i) A i B A i B In 38 45, find the multiplicative inverse of each of the following in a + bi form. 38. i i i i i i i pi In 46 60, write each quotient in a + bi form. 46. (8 4i) 4 ( i) 47. (2 4i) 4 ( 2 i) 48. (0 5i) 4 ( 2i) 49. (5 2 5i) 4 (3 2 i) i 4 2 2i i 2 6i 8 2i 52. 3i i 3 2 i i 5 2i 2 3i 8 6i i 3i 2i i 3i i p 2 i A i B A i B A i B 3 2 4i 7 i 5 i Appling Skills 6. Impedance is the resistance to the flow of current in an electric circuit measured in ohms. The impedance, Z, in a circuit is found b using the formula Z 5 V I where V is the voltage (measured in volts) and I is the current (measured in amperes). Find the impedance when V i volts and I 520.3i amperes. 62. Find the current that will flow when V i volts and Z 5.5 8i ohms using the formula. Z 5 V I

32 Comple Roots of a Quadratic Equation CMPLEX RTS F A QUADRATIC EQUATIN When we used the quadratic formula to find the roots of a quadratic equation, we found that some quadratic equations have no real roots. For eample, the quadratic equation has no real roots since the graph of the corresponding quadratic function f() , shown on the right, has no -intercepts. Can the roots of a quadratic equation be comple numbers? We will use the quadratic formula to find the f() roots of the equation For this equation, a 5, b 5 4, and c 5 5. The value of the discriminant is b 2 2 4ac ()(5) or 24 Therefore, the roots are not real numbers. 5 2b 6 "b2 2 4ac 2a "42 2 4()(5) 24 6!24 2() i i The roots are the comple numbers 22 i and 22 2 i. Therefore, ever quadratic equation with rational coefficients has two comple roots. Those roots ma be two real numbers. Comple roots that are real ma be rational or irrational and equal or unequal. Roots that are not real numbers are comple conjugates. Note that comple roots that are not real are often referred to as imaginar roots and comple roots of the form 0 bi 5 bi as pure imaginar roots. A quadratic equation with rational coefficients has roots that are pure imaginar onl if b 5 0 and b 2 2 4ac, 0. Let a 2 b c 5 0 be a quadratic equation and a, b, and c be rational numbers with a 0: The number When the discriminant The roots of the of -intercepts b 2 2 4ac is equation are of the function is:. 0 and a perfect square real, rational, and unequal 2. 0 and not a perfect square real, irrational, and unequal real, rational, and equal, 0 imaginar numbers 0

33 28 Quadratic Functions and Comple Numbers EXAMPLE Find the comple roots of the equation Solution How to Proceed () Write the equation in standard form: (2) Determine the values of a, b, and c: (3) Substitute the values of a, b, and c into the quadratic formula: (4) Perform the computation: (5) Write!22 in simplest form in terms of i: a 5, b 526, c b 6 "b2 2 4ac 2a 5 2(26) 6 "(26)2 2 4()(2) 2() 5 6 6! ! !4!3! i! i!3 Answer The roots are 3 i!3 and 3 2 i!3. EXAMPLE 2 For what values of c does the equation c 5 0 have imaginar roots? Solution How to Proceed () The equation has imaginar roots when b 2 2 4ac, 0. Substitute a 5 2, b 5 3: (2) Isolate the term in c: (3) Solve the inequalit. Dividing b a negative number reverses the order of the inequalit: 9 Answer The roots are imaginar when c. 8. b 2 2 4ac, (2)c, c, 0 28c, 29 28c c. 9 8

34 Sum and Product of the Roots of a Quadratic Equation 29 Eercises Writing About Mathematics. Emil said that when a and c are real numbers with the same sign and b 5 0, the roots of the equation a 2 b c 5 0 are pure imaginar. Do ou agree with Emil? Justif our answer. 2. Noah said that if a, b, and c are rational numbers and b 2 2 4ac, 0, then the roots of the equation a 2 b c 5 0 are comple conjugates. Do ou agree with Noah? Justif our answer. Developing Skills In 3 4, use the quadratic formula to find the imaginar roots of each equation = SUM AND PRDUCT F THE RTS F A QUADRATIC EQUATIN Writing a Quadratic Equation Given the Roots of the Equation We know that when a quadratic equation is in standard form, the polnomial can often be factored in order to find the roots. For eample, to find the roots of , we can factor the polnomial (2 2 3)( 2 )

35 220 Quadratic Functions and Comple Numbers EXAMPLE We can work backward on each of these steps to write a quadratic equation with a given pair of factors. For eample, write a quadratic equation whose roots are 3 and 25: () Let equal each of the roots: (2) Write equivalent equations with one side equal to 0: (3) If equals are multiplied b equals ( 2 3)( 5) 5 0(0) their products are equal: We can check to show that 3 and 25 are the roots of : Check for 5 3: Check for 525: (3) 2 2(3) 2 5 5? 0 (25) 2? 2(25) ? ? We can use the same procedure for the general case. Write the quadratic equation that has roots r and r 2 : () Let equal each of the roots: 5 r 5 r 2 (2) Write equivalent equations with 2 r r one side equal to 0: (3) If equals are multiplied b equals, ( 2 r 2 )( 2 r ) 5 0(0) their products are equal: 2 2 r 2 r 2 r r (r r 2 ) r r In the eample given above, r 5 3 and r The equation with these roots is: 2 2 (r r 2 ) r r [3 (25)] 3(25) !6 2!6 Write an equation with roots 3 and 3.!6 2!6 2 Solution r + r r r A!6B Q "6 R Q 2 "6 R