Complex Numbers. (x 1) (4x 8) n 2 4 x No real-number solutions. From the definition, it follows that i 2 1.

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1 7_Ch09_online 7// 0:7 AM Page 9-9. Comple Numbers 9- SECTION 9. OBJECTIVES Epress square roots of negative numbers in terms of i. Write comple numbers in a bi form. Add and subtract comple numbers. Multipl and divide comple numbers. Solve quadratic equations with complenumber solutions. The Language of Algebra For ears, mathematicians thought numbers like and were useless. In the 7th centur, René Descartes (9 0) referred to them as imaginar numbers. Toda the have important uses in electronics, engineering, and even biolog. Comple Numbers ARE YOU READY? The following problems review some basic skills that are needed when working with comple numbers.. Eplain wh is not a real number.. Add: ( ) ( 7). Subtract: ( ) ( 8). Multipl:. Multipl: 9( ) ( )(7 ). Solve: 0 Earlier in this chapter, we saw that some quadratic equations do not have real-number solutions. When we used the square root propert or the quadratic formula to solve them, we encountered the square root of a negative number. Section 9. Eample d Section 9. Eample Solve: n Solve: 0 n No real-number solutions No real-number solutions To solve such equations, we must define the square root of a negative number. Epress Square Roots of Negative Numbers in Terms of i. Recall that the square root of a negative number is not a real number. However, an epanded number sstem, called the comple number sstem, has been devised to give meaning to epressions such as and. To define comple numbers, we use a new tpe of number that is denoted b the letter i. The Number i The imaginar number i i is defined as From the definition, it follows that i. We can use etensions of the product and quotient rules for radicals to write the square root of a negative number as the product of a real number and. i EXAMPLE Write each epression in terms of i: a. b. c. 8 d. B Strateg We will write each radicand as the product of and a positive number. Then we will appl the appropriate rules for radicals. Wh We want our work to produce a factor of so that we can replace it with i.

2 7_Ch09_online 7// 0:7 AM Page 9-9- CHAPTER 9 Quadratic Equations Solution Notation Since it is eas to confuse i with i, we write i first so that it is clear that the i is not within the radical smbol. However, both i and i are correct. After factoring the radicand, use the product rule for radicals. a. i i Replace with i. b. i or i Replace with i. c. 8 i i or i d. After factoring the radicand, use the product and quotient rules for radicals. B 9 i or B i Self Check Write each epression in terms of i: a. 8 b. c. 8 d Now Tr Problems, 7, and Write Comple Numbers in a bi Form. The imaginar number i is used to define comple numbers. Comple Numbers A comple number is an number that can be written in the form a bi, where a and b are real numbers and i. Comple numbers of the form a bi, where b 0, are also called imaginar numbers. Comple numbers of the form bi are also called pure imaginar numbers. For a comple number written in the standard form a bi, we call the real number a the real part and the real number b the imaginar part. EXAMPLE Write each number in the form a bi: a. b. c. 8 Strateg To write each number in the form a bi, we will determine a and bi. Wh The standard form a bi of a comple number is composed of two parts, the real part a and the imaginar part b. Solution a. 0i The real part is and the imaginar part is 0. Success Tip Comple numbers are used to describe alternating electric current, the are used to model fluid flow in a pipe, and to represent strains on steel beams in skscrapers. b. To simplif, we need to write in terms of i: i Thus, 0 i. The real part is 0 and the imaginar part is. c. To simplif 8, we need to write in terms of i: 9 i Thus, 8 8 i. The real part is 8 and the imaginar part is. Self Check Write each number in the form a bi: a. b. c. Now Tr Problems and 9 8

3 7_Ch09_online 7// 0:7 AM Page 9-9. Comple Numbers 9- The following table shows the relationship between the real numbers, the imaginar numbers, and the comple numbers. Real numbers.7 Comple numbers p Imaginar numbers 9 7i i i i i Add and Subtract Comple Numbers. Adding and subtracting comple numbers is similar to adding and subtracting polnomials. Addition and Subtraction of Comple Numbers. To add comple numbers, add their real parts and add their imaginar parts.. To subtract comple numbers, add the opposite of the comple number being subtracted. EXAMPLE Find each sum or difference. Write each result in the form a bi. a. ( i) ( 8i) b. ( i) ( i) c. i ( i) Strateg To add the comple numbers, we will add their real parts and add their imaginar parts. To subtract the comple numbers, we will add the opposite of the comple number to be subtracted. Wh We perform the indicated operations as if the comple numbers were polnomials with i as a variable. Success Tip Solution a. i is not a variable, but it is helpful to think of it as one when adding, subtracting, and multipling. For eample: i i 7i 8 i i i i i i ( i) ( 8i) ( ) ( 8)i Add 0i The sum of the imaginar parts. The sum of the real parts. b. ( i) ( i) ( i) ( i) To find the opposite, change the sign of each term of i. the opposite [ ()] ( )i Add the real parts. Add the imaginar parts. 9 i c. i ( i) ( )i Add the imaginar parts. 7i Self Check Find the sum or difference. Write each result in the form a bi: a. ( i) ( i) b. ( i) ( i) c. 9 ( i) Now Tr Problems and

4 7_Ch09_online 7// 0:7 AM Page 9-9- CHAPTER 9 Quadratic Equations Multipl and Divide Comple Numbers. Multipling comple numbers is similar to multipling polnomials. EXAMPLE Find each product. Write each result in the form a bi. a. i( i) b. ( i)( i) c. ( i)( i) Strateg We will use the distributive propert or the FOIL method to find the products. Wh We perform the indicated operations as if the comple numbers were polnomials with i as a variable. Solution a. i( i) i i i Distribute the multiplication b i. 0i i 0i () 0i 0i Replace with. Write the result in the form a bi. F O I L b. ( i)( i) 8i 0i 0i 8i 0() Use the FOIL method. 8i 0i 8i. Replace i with. 8i 0 8 8i i Subtract: 0 8. F O I L c. ( i)( i) 9 i i i 9 () Use the FOIL method. Add: i i 0. Replace i with. 9 Written in the form a bi, the product is 0i. Self Check Now Tr Find the product. Write each result in the form a bi: a. i( 9i) b. ( i)( i) c. ( i)( i) Problems and In Eample c, we saw that the product of the imaginar numbers i and i is the real number. We call i and i comple conjugates of each other. Comple Conjugates The comple numbers a bi and a bi are called comple conjugates. Success Tip To find the conjugate of a comple number, write it in standard form and change the sign between the real and imaginar parts from to or from to. For eample, 7 i and 7 i are comple conjugates. i and i are comple conjugates. i and i are comple conjugates, because i 0 i and i 0 i. In general, the product of the comple number a bi and its comple conjugate a bi is the real number a b, as the following work shows: F O I L (a bi)(a bi) a abi abi b i a b () a b Use the FOIL method. Add: abi abi 0. Replace with. i

5 7_Ch09_online 7// 0:7 AM Page 9-9. Comple Numbers 9- We can use this fact when dividing b a comple number. The process that we use is similar to rationalizing denominators. EXAMPLE Write each quotient in the form a bi: a. b. 7 i The Language of Algebra Recall that the word conjugate was used in Chapter 8 when we rationalized the denominators of radical epressions such as. Strateg We will build each fraction b multipling it b a form of that uses the conjugate of the denominator. Wh This step produces a real number in the denominator so that the result can then be written in the form a bi. Solution a. We want to build a fraction equivalent to 7 i that does not have i in the denominator. To make the denominator, 7 i, a real number, we need to multipl it b its comple 7 i 7 i 7 i 7 i conjugate,. It follows that should be the form of that is used to build. 7 i 7 i 7 i 7 i i 9 i i 9 () i 9 i i 7 i To build an equivalent fraction, multipl b 7 i. To multipl the numerators, distribute the multiplication b. To multipl the denominators, find (7 i)(7 i). Replace with. The denominator no longer contains i. i Simplif the denominator. i i This notation represents the sum of two fractions i that have the common denominator : and. Write the result in the form a bi. i b. We can make the denominator of i a real number b multipling it b the comple conjugate of i, which is i. Caution A common mistake is to replace i with. Remember, i. B definition, i and i. Notation It is acceptable to use a bi as a substitute for the form a (b)i. In Eample b, we write i instead of ()i. i i i i i i i i i i () () i i i i i To build an equivalent fraction, multipl b i. To multipl the numerators, find ( i)( i). To multipl the denominators, find ( i)( i). Replace with. The denominator no longer contains i. i Simplif the numerator and the denominator. Write each term of the numerator over the denominator,. Simplif each fraction. Self Check Write each quotient in standard form: a. b. i i i Now Tr Problems 9 and

6 7_Ch09_online 7// 0:7 AM Page 9-9- CHAPTER 9 Quadratic Equations As the results of Eample show, we use the following rule to divide comple numbers. Division of Comple Numbers To divide comple numbers, multipl the numerator and denominator b the comple conjugate of the denominator. Solve Quadratic Equations with Comple-Number Solutions. We have seen that certain quadratic equations do not have real-number solutions. In the comple number sstem, all quadratic equations have solutions. We can write their solutions in the form a bi. EXAMPLE Solve each equation. Epress the solutions in the form a bi: a. 0 b. ( ) Strateg We will use the square root propert to solve each equation. Wh It is the fastest wa to solve equations of this form. Solution a. 0 This is the equation to solve. To isolate, subtract from both sides. Use the square root propert. Note that the radicand is negative. i Write in terms of i : i. To epress the solutions in the form a bi, we must write 0 for the real part, a. Thus, the solutions are 0 i and 0 i, or more simpl, 0 i. b. ( ) This is the equation to solve. i i Use the square root propert. Note that the radicand is negative. Write in terms of i: 9 i. To isolate, add to both sides. The solutions are i and i, or more simpl, i. Self Check Solve each equation. Epress the solutions in the form a bi: a b. (d ) 8 Now Tr Problems 7 and EXAMPLE 7 Solve t t 0. Epress the solutions in the form a bi. Strateg We use the quadratic formula to solve the equation. Wh t t does not factor. Solution t b b ac In the quadratic formula, replace with t. a t () () ()() () t 8 8 Substitute for a, for b, and for c. Evaluate the power and multipl within the radical. Multipl in the denominator.

7 7_Ch09_online 7// 0:7 AM Page Comple Numbers 9-7 t 8 t i 8 t ( i) t i Subtract within the radical: 8. Note that the radicand is negative. Write in terms of i: i Factor out the GCF from i and factor 8 as. Then remove the common factor :. This notation represents the sum and difference of two fractions that have the common denominator : i and. Writing each result as a comple number in the form a bi, the solutions are i and i or more simpl i Self Check 7 Solve a a 0. Epress the solutions in the form a bi. Now Tr Problem SECTION 9. STUDY SET VOCABULARY Fill in the blanks.. 9 i is an eample of a number. The part is 9 and the part is.. The number i is used to define comple numbers. CONCEPTS Fill in the blanks.. a. i b. i i i.. a. i i b. i i. The product of an comple number and its comple conjugate is a number. i 7. To write the quotient i as a comple number in standard form, we multipl it b Determine whether each statement is true or false. a. Ever real number is a comple number. b. 7i is an imaginar number. c. is a real number. d. In the comple number sstem, all quadratic equations have solutions. 0. Give the comple conjugate of each number. a. 9i b. 8 i c. i d. i NOTATION. Write each epression so it is clear that i is not within the radical smbol. a. 7i b. i i. Write in the form a bi. GUIDED PRACTICE Write each epression in terms of i. See Eample B 9 B Write each number in the form a bi. See Eample

8 7_Ch09_online 7// 0:7 AM Page CHAPTER 9 Quadratic Equations Perform the operations. Write all answers in the form a bi. See Eample.. ( i) ( i). ( i) ( 9i) 7. ( 8i) ( 9i) ( i)( i) 7. ( 8i) ( 7i) i(7 i). (7 i) ( i). (8 i) (7 i) 7. ( i) 9i 8. (0 i) 7i 9. ( 9i) 0. (8 9i) Perform the operations. Write all answers in the form a bi. See Eample.. ( i). 9( i). i( i). i(7 i) i i i i 79. (0 9i) ( i) 80. ( i) ( i) 8. ( i) 8. 7( i) 8. ( i) 8. ( i) Solve each equation. Write all solutions in the form a bi ( i)( i). ( i)( i) 87. ( ) 88. ( 9) ( i)( i) 8. ( i)( i) 89. b b t t 0 Write each quotient in the form a bi. See Eample i i. i i. 7 i i. i i. i i. i i. 7 i i Solve each equation. Write all solutions in the form a bi. See Eample d a APPLICATIONS 97. Electricit. In an AC (alternating current) circuit, if two sections are connected in series and have the same current in each section, the voltage is given b V V V. Find the total voltage in a given circuit if the voltages in the individual sections are V 0..97i and V 8..79i.. ( ).. ( ) 7. ( ) ( ) Electronics. The impedance Z in an AC (alternating current) circuit is a measure of how much the circuit impedes (hinders) the flow of current through it. The impedance is related to the voltage V and the current I b the formula Solve each equation. Write all solutions in the form a bi. See Eample V IZ If a circuit has a current of (0..0i) amps and an impedance of (0..0i) ohms, find the voltage TRY IT YOURSELF Perform the operations. Write all answers in the form a bi i 7 i 7. ( i) (9 i) 7. ( i) ( i) WRITING 99. What unusual situation discussed at the beginning of this section illustrated the need to define the square root of a negative number? 00. Eplain the difference between the opposite of a comple number and its conjugate. 0. What is an imaginar number? 0. In this section, we have seen that i. From our previous eperience in this course, what is unusual about that fact?

9 7_Ch09_online 7// 0:7 AM Page Graphing Quadratic Equations 9-9 REVIEW Rationalize the denominator CHALLENGE PROBLEMS Perform the operations. 07. i 08. i i i SECTION 9. OBJECTIVES Understand the vocabular used to describe parabolas. Find the intercepts of a parabola. Determine the verte of a parabola. Graph equations of the form a b c. Solve quadratic equations graphicall. Graphing Quadratic Equations ARE YOU READY? The following problems review some basic skills that are needed when graphing quadratic equations.. Solve: 0. Evaluate 8 for.. What is the coefficient of each term of the epression?. Approimate to the nearest tenth. In this section, we will combine our graphing skills with our equation-solving skills to graph quadratic equations in two variables. Understand the Vocabular Used to Describe Parabolas. Equations that can be written in the form a b c, where a 0, are called quadratic equations in two variables. Some eamples are 8 8 In Section., we graphed, a quadratic equation in two variables. To do this, we constructed a table of solutions, plotted points, and joined them with a smooth curve, called a parabola. The parabola opens upward, and the lowest point on the graph, called the verte, is the point (0, 0). If we fold the graph paper along the -ais, the two sides of the parabola match. We sa that the graph is smmetric about the -ais and we call the -ais the ais of smmetr. The Language of Algebra An ais of smmetr (or line of smmetr) divides a parabola into two matching sides. The sides are said to be mirror images of each other. (, ) 9 (, 9) (, ) (, ) 0 0 (0, 0) (, ) (, ) 9 (, 9) (, 9) Ais of smmetr (, ) (, ) (, ) (0, 0) Verte (, ) (, 9) = 9 8 7

10 7_Ch09_online 7// 0:7 AM Page CHAPTER 9 Quadratic Equations In Section., we also graphed, another quadratic equation in two variables. The resulting parabola opens downward, and the verte (in this case, the highest point on the graph) is the point (0, ). The ais of smmetr is the -ais. The Language of Algebra In the equation a b c, each value of determines eactl one value of. Therefore, the equation defines to be a function of and we could write ƒ() a b c. Your instructor ma ask ou to use the vocabular and notation of functions throughout this section. (, ) 7 0 (, 7) (, ) (, ) (0, ) (, ) (, ) 7 (, 7) (, ) (, 7) Verte (0, ) (, ) 7 = + (, ) (, ) (, 7) Ais of smmetr For the equation, the coefficient of the term is the positive number. For, the coefficient of the term is the negative number. These observations suggest the following fact. Graphs of Quadratic Equations The graph of a b c, where a 0, is a parabola. It opens upward when a 0 and downward when a 0. The Language of Algebra The word parabolic (pronounced par a BOL ic) means having the form of a parabola. For eample, the light bulb in most flashlights is surrounded b a parabolic reflecting mirror. Parabolic shapes can be seen in a wide variet of real-world settings. These shapes can be modeled b quadratic equations in two variables. The path of a stream of water Parabola The shape of a satellite antenna dish Parabola The path of a thrown object Parabola Find the Intercepts of a Parabola. When graphing quadratic equations, it is helpful to know the - and -intercepts of the parabola. To find the intercepts of a parabola, we use the same steps that we used to find the intercepts of the graphs of linear equations. Finding Intercepts To find the -intercept, substitute 0 for in the given equation and solve for. To find the -intercepts, substitute 0 for in the given equation and solve for.

11 7_Ch09_online 7// 0:7 AM Page 9-9. Graphing Quadratic Equations 9- EXAMPLE Find the - and -intercepts of the graph of. Strateg To find the -intercept of the graph, we will let 0 and find. To find the -intercepts of the graph, we will let 0 and solve the resulting equation for. Wh A point on the -ais has an -coordinate of 0. A point on the -ais has a -coordinate of 0. Solution We let 0 and evaluate the right side to find the -intercept. This is the given equation. 0 (0) Substitute 0 for. Evaluate the right side. The -intercept is (0, ). We note that the -coordinate of the -intercept is the same as the value of the constant term c on the right side of. Net, we let 0 and solve the resulting quadratic equation to find the -intercepts. This is the given equation. 0 Substitute 0 for. 0 ( )( ) Factor the trinomial. 0 or 0 Set each factor equal to 0. Since there are two solutions, the graph has two -intercepts: (, 0) and (, 0). Self Check Find the - and -intercepts of the graph of 8. Now Tr Problem 7 -intercept Same length Verte -intercept Determine the Verte of a Parabola. It is usuall easier to graph a quadratic equation when we know the coordinates of the verte of its graph. Because of smmetr, if a parabola has two -intercepts, the -coordinate of the verte is eactl midwa between them. We can use this fact to derive a formula to find the verte of a parabola. In general, if a parabola has two -intercepts, the can be found b solving 0 a b c for. We can use the quadratic formula to find the solutions. The are b b ac and b b ac a a Thus, the parabola s -intercepts are and b b ac b b ac a, 0 a, 0. Since the -value of the verte of a parabola is halfwa between the two -intercepts, we can find this value b finding the average, or of the sum of the -coordinates of the -intercepts. ab b ac a ab b ac (b) b ac b a ab a b b a b b ac b a Add the numerators and keep the common denominator. Simplif: b (b) b and b ac b ac 0. Simplif the fraction within the parentheses. This result is true even if the graph has no -intercepts.

12 7_Ch09_online 7// 0:7 AM Page 9-9- CHAPTER 9 Quadratic Equations Finding the Verte of a Parabola The graph of the quadratic equation a b c is a parabola whose verte has an b b -coordinate of a. To find the -coordinate of the verte, substitute a for into the equation and find. EXAMPLE Find the verte of the graph of. Strateg We will compare the given equation to the general form a b c to identif a and b. Wh To use the verte formula, we need to know a and b. Solution From the following diagram, we see that a, b, and c. Success Tip An eas wa to remember the verte formula is to note that b a is part of the quadratic formula: b b ac a a b c To find the -coordinate of the verte, we substitute the values for a and b into the formula b a. b a () () The -coordinate of the verte is. To find the -coordinate, we substitute for in the original equation. () The verte of the parabola is (, ). This is the given equation. Substitute for. Evaluate the right side. Self Check Find the verte of the graph of 8. Now Tr Problem Graph Equations of the Form a b c. Much can be determined about the graph of a b c from the coefficients a, b, and c. We can use this information to help graph the equation. Graphing a Quadratic Equation a b c. Test for opening upward/downward Determine whether the parabola opens upward or downward. If a 0, the graph opens upward. If a 0, the graph opens downward.. Find the verte/ais of smmetr The -coordinate of the verte of the parabola is b b a. To find the -coordinate, substitute a for into the equation and find. The ais of smmetr is the vertical line passing through the verte.. Find the intercepts To find the -intercept, substitute 0 for in the given equation and solve for. The result will be c. Thus, the -intercept is (0, c). To find the -intercepts (if an), substitute 0 for and solve the resulting quadratic equation a b c 0. If no real-number solutions eist, the graph has no -intercepts.. Plotting points/using smmetr To find two more points on the graph, select a convenient value for and find the corresponding value of. Plot that point and its mirror image on the opposite side of the ais of smmetr.. Draw the parabola Draw a smooth curve through the located points.

13 7_Ch09_online 7// 0:7 AM Page 9-9. Graphing Quadratic Equations 9- EXAMPLE Graph: Strateg We will use the five-step procedure described on the previous page to sketch the graph of the equation. Wh This strateg is usuall faster than making a table of solutions and plotting points. Solution Upward/downward: The equation is in the form a b c, with a, b, Success Tip and c. Since a 0, the parabola opens upward. When graphing, remember that Verte/ais of smmetr: In Eample, we found that the verte of the graph is (, ). an point on a parabola to the right of the ais of smmetr ields a second point to the left of the The ais of smmetr will be the vertical line passing through (, ). See part (a) of the figure below. ais of smmetr, and vice versa. Intercepts: In Eample, we found that the -intercept is (0, ) and -intercepts are Think of it as two-for-one. (, 0) and (, 0). If the point (0, ), which is unit to the left of the ais of smmetr, is on the graph, the point (, ), which is unit to the right of the ais of smmetr, is also on the graph. See part (a). Success Tip Plotting points/using smmetr: It would be helpful to locate two more points on the graph. To find a solution of, we select a convenient value for, sa, To find additional points on the graph, select values of that are and find the corresponding value of. close to the -coordinate of the verte. () () This is the equation to graph. Substitute for. Evaluate the right side. Thus, the point (, ) lies on the parabola. If the point (, ), which is units to the left (, ) of the ais of smmetr, is on the graph, the point (, ), which is units to the right of the ais of smmetr, is also on the graph. See part (a). (, ) Draw a smooth curve through the points: The completed graph of is shown in part (b). (, ) (, ) From table Ais of smmetr (, 0) (, 0) -intercept -intercept (0, ) (, ) From -intercept (, ) smmetr Verte (, 0) (, 0) From smmetr (, ) (, ) (0, ) (, ) (, ) = (a) (b) Self Check Use our results from Self Checks and to help graph 8. Now Tr Problem EXAMPLE Graph: Strateg 8 8 We will follow the five-step procedure to sketch the graph of the equation. Wh This strateg is usuall faster than making a table of solutions and plotting points.

14 7_Ch09_online 7// 0:7 AM Page 9-9- CHAPTER 9 Quadratic Equations Solution Upward/downward: The equation is in the form a b c, with a, Success Tip b 8, and c 8. Since a 0, the parabola opens downward. Verte/ais of smmetr: To find the -coordinate of the verte, we substitute for a The most important point to find when graphing a quadratic and 8 for b into the formula b a. equation in two variables is the verte. b a (8) () The -coordinate of the verte is. To find the -coordinate, we substitute for in the original equation and find. 8 8 This is the equation to graph. () 8() 8 Substitute for. 0 Evaluate the right side. The verte of the parabola is the point (, 0). The ais of smmetr is the vertical line passing through (, 0). See part (a) of the figure. Intercepts: Since c 8, the -intercept of the parabola is (0, 8). The point (, 8), which is units to the left of the ais of smmetr, must also be on the graph. See part (a). To find the -intercepts, we let 0 and solve the resulting quadratic equation. Success Tip 8 8 This is the equation to graph. To solve 0 8 8, Substitute 0 for. instead of dividing both sides b 0 0 Divide both sides b : 8 8., we could factor out the common factor and proceed 0 ( )( ) Factor the trinomial. as follows: 0 or 0 Set each factor equal to ( ) 0 ( )( ) Since the solutions are the same, the graph has onl one -intercept: (, 0). This point is the verte of the parabola and has alread been plotted. 8 8 Plotting points/using smmetr: It would be helpful to know two more points on the graph. To find a solution of 8 8, we select a convenient value for, sa (, ), and find the corresponding value for. (, ) 8 8 () 8() 8 This is the equation to graph. Substitute for. Evaluate the right side. Thus, the point (, ) lies on the parabola. We plot (, ) and then use smmetr to determine that (, ) is also on the graph. See part (a). Draw a smooth curve through the points: The completed graph of 8 8 is shown in part (b). Verte -intercept (, ) (, 0) (, ) From smmetr (, 0) (, ) (, ) = 8 8 (, 8) From smmetr (0, 8) -intercept (, 8) 9 (0, 8) (a) (b)

15 7_Ch09_online 7// 0:7 AM Page 9-9. Graphing Quadratic Equations 9- Self Check Graph: Now Tr Problem 9 EXAMPLE Graph: Strateg We will follow the five-step procedure to sketch the graph of the equation. Wh This strateg is usuall faster than making a table of solutions and plotting points. Solution Upward/downward: The equation is in the form a b c, with a, b, and c. Since a 0, the parabola opens upward. Verte/ais of smmetr: To find the -coordinate of the verte, we substitute for a and for b into the formula b a. b a () The -coordinate of the verte is. To find the -coordinate, we substitute for in the original equation and find. () () This is the equation to graph. Substitute for. Evaluate the right side. The verte of the parabola is the point (, ). The ais of smmetr is the vertical line passing through (, ). See part (a) of the figure on the net page. Intercepts: Since c, the -intercept of the parabola is (0, ). The point (, ), which is one unit to the left of the ais of smmetr, must also be on the graph. See part (a). To find the -intercepts, we let 0 and solve the resulting quadratic equation. 0 This is the equation to graph. Substitute 0 for. Since does not factor, we will use the quadratic formula to solve for. ()() () ( ) In the quadratic formula, substitute for a, for b, and for c. Evaluate the right side. Simplif the radical:. Factor out the GCF,, from the terms in the numerator. Then simplif b removing the common factor of in the numerator and denominator. The -intercepts of the graph are, 0 and, 0. To help to locate their positions on the graph, we can use a calculator to approimate the two irrational numbers. See part (a) on the net page. 0.7 and.7

16 7_Ch09_online 7// 0:7 AM Page 9-9- CHAPTER 9 Quadratic Equations Plotting points/using smmetr: To find two more points on the graph, we let, substitute for in, and find that is. We plot (, ) and then use (, ) smmetr to determine that (, ) is also on the graph. See part (a). (, ) Draw a smooth curve through the points: The completed graph of is shown in part (b). Success Tip To plot the -intercepts, we use the approimations (0.7, 0) and (.7, 0) to locate their position on the -ais. However, we label the points on the graph using the eact values:, 0 and, 0. (, ) From smmetr (, 0) -intercept Ais of smmetr (, ) From table ( +, 0) -intercept (, ) (0, ) From -intercept smmetr (, ) Verte (, ) (, 0) (, ) = + (, ) (0, ) (, ) ( +, 0) (a) (b) Self Check Graph: Now Tr Problem Solve Quadratic Equations Graphicall. The number of distinct -intercepts of the graph of a b c is the same as the number of distinct real-number solutions of a b c 0. For eample, the graph of in part (a) of the following figure has two -intercepts, and 0 has two real-number solutions. In part (b), the graph has one -intercept, and the corresponding equation has one real-number solution. In part (c), the graph does not have an -intercept, and the corresponding equation does not have an real-number solutions. Note that the solutions of each equation are given b the -coordinates of the -intercepts of each respective graph. (, 0) (, 0) = + + = 0 has two solutions, and. (a) (, 0) 7 = = 0 has one repeated solution,. (b) 7 7 = + + = 0 has no real-number solutions. (c)

17 7_Ch09_online 7// 0:7 AM Page Graphing Quadratic Equations 9-7 SECTION 9. STUDY SET VOCABULARY Fill in the blanks.. is a equation in two variables. Its graph is a cup-shaped figure called a.. The lowest point on a parabola that opens upward, and the highest point on a parabola that opens downward, is called the of the parabola.. Points where a parabola intersects the -ais are called the - of the graph and the point where a parabola intersects the -ais is called the - of the graph.. The vertical line that splits the graph of a parabola into two identical parts is called the ais of. CONCEPTS Fill in the blanks.. The graph of a b c opens downward when a 0 and upward when a.. The graph of a b c is a parabola whose verte has an -coordinate given b. 7. a. To find the -intercepts of a graph, substitute for in the given equation and solve for. b. To find the -intercepts of a graph, substitute 0 for in the given equation and solve for. 8. (, ) (, ) 9. a. What do we call the curve shown in the graph? b. What are the -intercepts of the graph? c. What is the -intercept of the graph? d. What is the verte? e. Draw the ais of smmetr on the graph.. Sketch the graph of a quadratic equation using the given facts about its graph. Opens upward -intercept: (0, 8) Verte: (, ) -intercepts: (, 0), (, 0). Eamine the graph of. How man real-number solutions does the equation 0 have? Find them. NOTATION. Consider the equation 8. a. What are a, b, and c? b b. Find a.. Evaluate: () () GUIDED PRACTICE (, ) (, ). Eamine the graph of. How man real-number solutions does the equation 0 have? Find them. Find the - and -intercepts of the graph of the quadratic equation. See Eample = + 7 = Does the graph of each quadratic equation open upward or downward? a. b.. The verte of a parabola is (, ), its -intercept is (0, ), and it passes through the point (, ). Draw the ais of smmetr and use it to help determine two other points on the parabola Find the verte of the graph of each quadratic equation. See Eample Graph each quadratic equation b finding the verte, the - and -intercepts, and the ais of smmetr of its graph. See Eamples and

18 7_Ch09_online 7// 0:7 AM Page CHAPTER 9 Quadratic Equations Graph each quadratic equation b finding the verte, the - and -intercepts, and the ais of smmetr of its graph. See Eample Health Department. The number of cases of flu seen b doctors at a count health clinic each week during a 0-week period is described b the graph. Write a brief summar report about the flu outbreak. What important piece of information does the verte give? Cases of the flu Week. Trampolines. The graph shows how far a trampolinist is from the ground (in relation to time) as she bounds into the air and then falls back down to the trampoline. a. How man feet above the ground is she second after bounding upward? b. When is she 9 feet above the ground? c. What is the maimum number of feet above the ground she gets? When does this occur? APPLICATIONS. Biolog. Draw an ais of smmetr over the sketch of the butterfl.. Crossword Puzzles. Darken the appropriate squares to the right of the dashed blue line so that the puzzle has smmetr with respect to that line. ft Height above the ground (ft) ais of smmetr. Cost Analsis. A compan has found that when it assembles carburetors in a production run, the manufacturing cost of $ per carburetor is given b the graph below. What important piece of information does the verte give?.0. Time (sec).0. Track and Field. Sketch the parabolic path traveled b the long-jumper s center of gravit from the take-off board to the landing. Let the -ais represent the ground. Cost per carburetor ($) ft Take-off board Carburetors assembled ft Landing

19 7_Ch09_online 7// 0:7 AM Page Graphing Quadratic Equations 9-9 WRITING 7. A mirror is held against the -ais of the graph of a quadratic equation. What fact about parabolas does this illustrate? REVIEW Simplif each epression z z z. 7 z CHALLENGE PROBLEMS 7. Graph and use the graph to solve the quadratic equation. 8. Use the eample of a stream of water from a drinking fountain to eplain the concept of the verte of a parabola. 9. Eplain wh the -intercept of the graph of the quadratic equation a b c is (0, c). 0. Eplain how to determine from its equation whether the graph of a parabola opens upward or downward.. Is it possible for the graph of a parabola with equation of the form a b c not to have a -intercept? Eplain.. Sketch the graphs of parabolas with zero, one, and two - intercepts. Can the graph of a quadratic equation of the form a b c have more than two -intercepts? Eplain wh or wh not. 8. What quadratic equation is graphed here?

20 7_Ch09_online 7// 0:7 AM Page CHAPTER 9 Quadratic Equations SECTION 9. Comple Numbers DEFINITIONS AND CONCEPTS EXAMPLES The imaginar number i is defined as Write each epression in terms of i : i From the definition, it follows that i. A comple number is an number that can be written in the form a bi, where a and b are real numbers and i. We call a the real part and b the imaginar part. If b 0, a bi is also an imaginar number i 0 i 0i i or i Show that each number is a comple number b writing it in the form a bi i 0 is the real part and 0 is the imaginar part. i 0 i 0 is the real part and is the imaginar part. i is the real part and is the imaginar part. Adding and subtracting comple numbers is similar to adding and subtracting polnomials. To add two comple numbers, add their real parts and add their imaginar parts. Add: ( i) ( 7i) ( ) ( 7)i 8 i Add the real parts. Add the imaginar parts. To subtract two comple numbers, add the opposite of the comple number being subtracted. Subtract: ( i) ( 7i) ( i) ( 7i) ( ) [ (7)]i i Add the opposite of 7i. Add the real parts. Add the imaginar parts. Multipling comple numbers is similar to multipling polnomials. Find each product: 7i( 9i) 8i i ( i)( i) 8 i i 9i 8i () 8 i 9() 8i 8 i 9 8i 7 i The comple numbers a bi and a bi are The comple numbers i and i are comple conjugates. called comple conjugates. To write the quotient of two comple numbers in the form a bi, multipl the numerator and denominator b the comple conjugate of the denominator. The process is similar to rationalizing denominators. This process is similar to rationalizing two-term radical denominators. Write each quotient in the form : i 8i 9() 8i 9 8i 8 i a bi i i i i i i i i i 8i i i i 9i i i () () i i

21 7_Ch09_online 7// 0:7 AM Page 9- CHAPTER 9 Summar & Review 9- Some quadratic equations have comple solutions that are imaginar numbers. Use the quadratic formula to solve: Here, a, b, and c. p b b ac a p ()() () p p i p i p p 0 In the quadratic formula, replace with p. Substitute for a, for b, and for c. Evaluate the power and multipl within the radical. Multipl in the denominator. Write in terms of i. Write the solutions in the form a bi. REVIEW EXERCISES Write each epression in terms of i.. 9i. i Perform the operations. Write all answers in the form a bi B 9 7. ( i) ( i) 8. (7 i) ( i) 9. i( i) 0. ( i)( i). Complete the diagram. i Comple numbers.. i i. Determine whether each statement is true or false. a. Ever real number is a comple number. b. i is an imaginar number. c. is a real number. d. i is a real number. Give the comple conjugate of each number.. i. 7i Solve each equation. Write all solutions in the form a bi (p ). (q ) SECTION 9. Graphing Quadratic Equations DEFINITIONS AND CONCEPTS The verte of a parabola is the lowest (or highest) point on the parabola. A vertical line through the verte of a parabola that opens upward or downward is called its ais of smmetr. EXAMPLES A parabola Ais of smmetr Verte (, )

22 7_Ch09_online 7// 0:7 AM Page 9-9- CHAPTER 9 Quadratic Equations Equations that can be written in the form a b c, where a 0, are called quadratic equations in two variables. The graph of a b c is a parabola. Much can be determined about the graph from the coefficients a, b, and c. The parabola opens upward when a 0 and downward when a 0. The -coordinate of the verte of the parabola is b a. To find the -coordinate of the verte, b substitute a for in the equation of the parabola and find. To find the -intercept, substitute 0 for in the given equation and solve for. To find the -intercepts, substitute 0 for in the given equation and solve for. The number of distinct -intercepts of the graph of a quadratic equation a b c is the same as the number of distinct real-number solutions of a b c 0. Graph: Upward/downward: The equation is in the form a b c, with a, b, and c 8. Since a 0, the parabola opens upward. Verte/ais of smmetr: b a () The -coordinate of the verte is: 8 () () The verte of the parabola is (, ). The -coordinate of the verte is The equation to graph. Substitute for. Intercepts: Since c 8, the -intercept of the parabola is (0, 8). The point (, 8), which is units to the left of the ais of smmetr, must also be on the graph. To find the -intercepts of the graph of 8, we set 0 and solve for ( )( ) 0 or 0 The -intercepts of the graph are (, 0) and (, 0). Plotting points/using smmetr: To locate two more points on the graph, we let and find the corresponding value of. () () 8 Thus, the point (, ) lies on the parabola. We use smmetr to determine that (, ) is also on the graph. Draw a smooth curve through the points: The completed graph of 8 is shown here. (, 8) -intercept 7 (0, 8) (, ) -intercept (, 0) Ais of smmetr (, ) -intercept Verte (, 0) (, ) 7 8 Since the graph of 8 (shown above) has two -intercepts, (, 0) and (, 0), the equation 8 0 has two distinct real-number solutions, and.

23 7_Ch09_online 7// 0:7 AM Page 9- CHAPTER 9 Test 9- REVIEW EXERCISES 9. Refer to the figure. a. What are the -intercepts of the parabola? b. What is the -intercept of the parabola? c. What is the verte of the parabola? d. Draw the ais of smmetr of the parabola on the graph. 70. The point (0, ) lies on the parabola graphed above. Use smmetr to determine the coordinates of another point that lies on the parabola. Find the verte of the graph of each quadratic equation and tell in which direction the parabola opens. Do not draw the graph Find the - and -intercepts of the graph of each quadratic equation Graph each quadratic equation b finding the verte, the - and -intercepts, and the ais of smmetr of its graph The graphs of three quadratic equations in two variables are shown. Fill in the blanks. 80. Manufacturing. What important information can be obtained from the verte of the parabola in the graph below? = + 7 = = + 0 has 8 0 has 0 has real-number solution(s). repeated real-number solution. real-number solution(s). Give the solution(s): Give the solution(s): 7 Profit ($,000s) Units sold (00s) 9 CHAPTER TEST. Fill in the blanks. a. A equation can be written in the form a b c 0, where a, b, and c represent real numbers and a 0. b. 8 is a perfect- trinomial because 8 ( ). c. When we add to 0, we sa we have the square on 0. d. We read as three or the square root of two. e. The coefficient of 8 9 is and the term is 9.. Write the statement or using double-sign notation. Solve each equation b the square root method.. 7. r 8 0. ( ) t Eplain wh the equation m 9 0 has no real-number solutions. 0. Check to determine whether is a solution of n 0. Complete the square and factor the resulting perfect-square trinomial..... c 7c a a. Complete the square to solve a a 0. Give the eact solutions and then approimate them to the nearest hundredth.

24 7_Ch09_online 7// 0:7 AM Page 9-9- CHAPTER 9 Quadratic Equations Use the most efficient method to solve each equation.. Complete the square to solve a a. 7. Complete the square to solve m m Complete the square to solve: (b ). u 0. n n 0 Use the quadratic formula to solve each equation. Write each epression in terms of i n n 0. 7t t. 8 Perform the operations. Write all answers in the form a bi.. Solve 0 using the quadratic formula. Give the eact solutions, and then approimate them to the nearest hundredth.. Check to determine whether is a solution of 0.. Archer. The area of a circular archer target is,0 cm. What is the radius of the target? Round to the nearest centimeter.. (8 i) ( 7 i). ( i) ( 9i) 7. ( i)( i) 8. Solve each equation. Write all solutions in the form a bi n 8. Geometr. The hpotenuse of a right triangle is 8 feet long. One leg is feet longer than the other. Find the lengths of the legs. Round to the nearest tenth. 0. (a ). 0. Advertising. When a business runs advertisements per week on television, the number of air conditioners it sells is given b the graph in the net column. What important information can be obtained from the verte? Number of AC units sold that week AP Photo Number of TV ads run during the week. Fill in the blanks: The graph of a b c opens downward when a 0 and upward when a 0. emin kuliev/shutterstock.com. St. Louis. On October 8, 9, workers topped out the final section of the Gatewa Arch in St. Louis, Missouri. It is the tallest national monument in the United States at 0 feet. If a worker dropped a tool from that height, how long would it take to reach the ground? Round to the nearest tenth. 7. New York Cit. The rectangular Samsung sign in Times Square is a full color LED screen that has an area of, ft. Its height is 7 feet less than twice its width. Find the width and height of the sign. i i Graph each quadratic equation b finding the verte, the - and -intercepts, and the ais of smmetr of its graph... 7

25 7_Ch09_online 7// 0:7 AM Page 9- CHAPTER 9 Cumulative Review 9- CUMULATIVE REVIEW Chapters 9. Determine whether each statement is true or false. [Section.] a. Ever rational number can be written as a ratio of two integers. b. The set of real numbers corresponds to all points on the number line. c. The whole numbers and their opposites form the set of integers.. Driving Safet. In cold-weather climates, salt is spread on roads to keep snow and ice from bonding to the pavement. This allows snowplows to remove built-up snow quickl. According to the graph, when is the accident rate the highest? [Section.] Accident Rate Before and After Salt Spreading Accident rate (per 0 million vehicle kilometers) Spreading time Time (in hours) Based on data from a stud done in Europe b the Salt Institute. Evaluate: [7 (9)] [Section.7]. Evaluate: [Section.7]. Evaluate ( a) ( b) for,, a, and b. [Section.8]. Simplif: p (p 9) p [Section.9] 7. Solve k 0 and check the result. [Section.] 8. Solve (a ) a and check the result. [Section.] 9. Loose Change. The Coinstar machines that are in man grocer stores count unsorted coins and print out a voucher that can be echanged for cash at the checkout stand. However, to use this service, a processing fee is charged. If a bo turned in a jar of coins worth $0 and received a voucher for $., what was the processing fee (epressed as a percent) charged b Coinstar? [Section.] 0. Solve T r t for r. [Section.]. Selling a Home. At what price should a home be listed if the owner wants to make $0,000 on its sale after paing a % real estate commission? [Section.]. Business Loans. Last ear, a women s professional organization made two small-business loans totaling $8,000 to oung women beginning their own businesses. The mone was lent at 7% and 0% simple interest rates. If the annual income the organization received from these loans was $,0, what was each loan amount? [Section.] Number of viewers. Solve 7 and graph the solution set. Then use interval notation to describe the solution. [Section.7]. Check to determine whether (, ) is a solution of. [Section.] Graph each equation or inequalit.. [Section.]. [Section.] 7. [Section.] 8. [Section.7] 9. Find the slope of the line passing through (, ) and (, 8). [Section.] 0. TV News. The line graph in red below approimates the evening news viewership on all networks for the ears Find the rate of decrease over this period of time. [Section.],00,000,900, Year 09 Source: The State of the News Media, 00. What is the slope of the line defined b? [Section.]. Write the equation of the line whose graph has slope and -intercept (0, ). [Section.]. Are the graphs of 9 and 0 parallel, perpendicular, or neither? [Section.]. Write the equation of the line whose graph has slope and passes through the point (8, ). Write the equation in slope intercept form. [Section.]. Graph the line passing through (, ) and having slope. [Section.]. If ƒ() 8, find ƒ(). [Section.8] 7. Find the domain and range of the relation: {(, 8), (, ), (, ), (, 8)} [Section.8] 8. Is this the graph of a function? [Section.8]

26 7_Ch09_online 7// 0:7 AM Page 9-9- CHAPTER 9 Quadratic Equations 9. Solve using the graphing method. [Section.] e 0. Solve using the substitution method. e [Section.]. Solve using the elimination (addition) method. s t μ s 8 t [Section.]. Aviation. With the wind, a plane can fl,000 miles in hours. Against the wind, the trip takes hours. Find the airspeed of the plane (the speed in still air). Use two variables to solve this problem. [Section.]. Miing Cand. How man pounds of each cand must be mied to obtain 8 pounds of cand that would be worth $.0 per pound? Use two variables to solve this problem. [Section.] Perform the operations.. ( 8) ( 8) [Section.]. (a )(a ) [Section.]. (b )(b ) [Section.]. ( )( ) [Section.] 7. ( ) [Section.7] 8. (9m )(9m ) [Section.7] a b 9a b ab 9. [Section.8] a b 0. [Section.8] Factor each epression completel.. a a b ab. a a [Section.] [Section.] [Section.] [Section.]. t. b [Section.] [Section.]. Solve the sstem of linear inequalities. 7 e [Section.] Simplif each epression. Write each answer without using parentheses or negative eponents.. [Section.]. a b ( ) [Section.] a b a z 0a a b a a 0 7 [Section.] [Section.] 9. Five-Card Poker. The odds against being dealt the hand shown are about. 0 to. Epress. 0 using standard notation. [Section.] Solve each equation b factoring [Section.7] [Section.7] 9. Geometr. The triangle shown has an area of. square inches. Find its height. [Section.8] + 0. For what value is 8 undefined? [Section 7.] Simplif each epression.. 7 a. 8 a [Section 7.] [Section 7.] Perform the operations and simplif when possible.. [Section 7.] 9 0. [Section 7.] s s s s s.. [Section 7.] [Section 7.] 0. Write in scientific notation. [Section.]. Graph: [Section.]. Write a polnomial that represents the perimeter of the rectangle. [Section.] + Simplif each comple fraction. 9m 7 m m m 8 [Section 7.] [Section 7.]

27 7_Ch09_online 7// 0:7 AM Page 9-7 CHAPTER 9 Cumulative Review 9-7 Solve each equation. 9. p [Section 7.] p p [Section 7.] q q q q 7. Solve the formula a b for a. [Section 7.] 7. Roofing. A homeowner estimates that it will take him 7 das to roof his house. A professional roofer estimates that he can roof the house in das. How long will it take if the homeowner helps the roofer? [Section 7.7] 7. Losing Weight. If a person cuts his or her dail calorie intake b 00, it will take 0 das for that person to lose 0 pounds. How long will it take for the person to lose pounds? [Section 7.8] 7. DABC and DDEF are similar D triangles. Find. [Section 7.8] A B 7. Suppose w varies directl as. If w. when, find w when 0. [Section 7.9] 7. Gears. The speed of a gear varies inversel with the number of teeth. If a gear with 0 teeth makes revolutions per second, how man revolutions per second will a gear with teeth make? [Section 7.9] Simplif each radical epression. All variables represent positive numbers b [Section 8.] [Section 8.] Perform the indicated operation [Section 8.] [Section 8.] Rationalize the denominator [Section 8.] 8. [Section 8.] 0 a Solve each equation t 7 t [Section 8.] [Section 8.] E F C 9. Storage Cubes. The diagonal distance across the face of each of the stacking cubes is inches. What is the height of the entire storage arrangement? Round to the nearest tenth of an inch. [Section 9.] 9. Solve 8 0 b completing the square. [Section 9.] 9. Solve 0 using the quadratic formula. Give the eact solutions, and then approimate each to the nearest hundredth. [Section 9.] 9. Quilts. According to the Guinness Book of World Records 998, the world s largest quilt was made b the Seniors Association of Saskatchewan, Canada, in 99. If the length of the rectangular quilt is feet less than twice its width and it has an area of,8 ft, find its width and length. [Section 9.] Write each epression in terms of i [Section 9.] 9. [Section 9.] Perform the operations. Epress each answer in the form a bi. 97. ( i) ( i) 98. (7 i) (9 i) [Section 9.] [Section 9.] 99. ( i)( i) 00. i i [Section 9.] [Section 9.] Solve each equation. Epress the solutions in the form a bi [Section 9.] [Section 9.] 0. Graph the quadratic equation 8. Find the verte, the - and -intercepts, and the ais of smmetr of the graph. [Section 9.] 0. Power Output. The graph shows the power output (in horsepower, hp) of a certain engine for various engine speeds (in revolutions per minute, rpm). For what engine speed does the power output reach a maimum? [Section 9.] in. Simplif each radical epression. All variables represent positive numbers. 7m 8. [Section 8.] 8. [Section 8.] B 8n Evaluate each epression. 87. [Section 8.] 88. [Section 8.] / Solve each equation. (8) / 89. t ( ) 7 0 [Section 9.] [Section 9.] Maimum power output (hp) ,000,000,000,000,000 Speed of engine (rpm)