7_Ch09_online 7// 0:7 AM Page 9-0 9-0 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. 00 00 7 7 00 7 i 0 i 0i i or i Show that each number is a comple number b writing it in the form a bi. 0 0 0i 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 9i 8i 9() 8i 9 8i 8 i a bi i i i i i i i i i i i i i () () i i i
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 p i 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 7. 8. 8 Perform the operations. Write all answers in the form a bi. 9. 9 0. 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.. 9 0.. (p ). (q ) 7. 8. 0 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 (, )
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 () () 8 9 8 8 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 8 0 ( )( ) 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 8 7 8 (, ) -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.
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 b. What is the -intercept of the c. What is the verte of the 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. 7. 7 7. 8 Find the - and -intercepts of the graph of each quadratic equation. 7. 7. Graph each quadratic equation b finding the verte, the - and -intercepts, and the ais of smmetr of its graph. 7. 7. 77. 78. 79. 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 = + + 8 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) 8 0 8 7 8 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. ( ). 0 7. t 8. 9 9. 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.
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 0 0. 8. Complete the square to solve: 9. 0. (b ). u 0. n n 0 Use the quadratic formula to solve each equation. Write each epression in terms of i. 9. 0 0.. 00. 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. 9. 00 0. 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 8 9 7 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