It s Not Complex Just Its Solutions Are Complex!
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1 It s Not Comple Just Its Solutions Are Comple! Solving Quadratics with Comple Solutions 15.5 Learning Goals In this lesson, ou will: Calculate comple roots of quadratic equations and comple zeros of quadratic functions. Interpret comple roots of quadratic equations and comple zeros of quadratic functions. Determine whether a function has comple solutions from a graph and from an equation in radical form. Determine the number of roots of a quadratic equation from a graph and from an equation in radical form. Ke Terms imaginar roots imaginar zeros So, all this talk about real and imaginar numbers can appear to be quite comple no pun intended but reall, it s a matter of determining where the -intercepts occur or if the even occur. You ve successfull gone through this course determining roots, intercepts, and zeros of quadratics; and with the eception of one question, all the solutions ou have encountered so far have been real. So, don t be worried: imaginar numbers are nothing to fret about the reall aren t that comple! 1113
2 15 Problem 1 Does It Intersect the? 1. Consider the two quadratic functions and their graphs shown. f() g() a. List all the ke characteristics ou know about f(). Be sure to include the number of zeros, the -intercept(s), the -intercept, the ais of smmetr, and the verte. b. List all the ke characteristics ou know about g(). Be sure to include the number of zeros, the -intercept(s), the -intercept, the ais of smmetr, and the verte. c. Compare f() and g(). What do the have in common? What is different about the two functions? 111 Chapter 15 Real Number Sstems
3 . Consider the two quadratic functions and their graphs shown. c() 5 1 d() a. List all the ke characteristics ou know about c(). Be sure to include the number of zeros, the -intercept(s), the -intercept, the ais of smmetr, and the verte. b. List all the ke characteristics ou know about d(). Be sure to include the number of zeros, the -intercept(s), the -intercept, the ais of smmetr, and the verte. c. Compare c() and d(). What do the have in common? What is different about the two functions? 15.5 Solving Quadratics with Comple Solutions 1115
4 15 3. Consider the two quadratic functions and their graphs shown. p() 5 1 q() a. List all the ke characteristics ou know about p(). Be sure to include the number of zeros, the -intercept(s), the -intercept, the ais of smmetr, and the verte. b. List all the ke characteristics ou know about q(). Be sure to include the number of zeros, the -intercept(s), the -intercept, the ais of smmetr, and the verte. c. Compare p() and q(). What do the have in common? What is different about the two functions? 111 Chapter 15 Real Number Sstems
5 Problem I See! No -intercept Means Imaginar! Before learning about the set of comple numbers, ou probabl would have said that the quadratic equations in Problem 1 Question 3 had no real solutions. Now ou know that the have imaginar solutions. Functions and equations that have imaginar solutions have imaginar roots or imaginar zeros, which are the solutions. 1. How can ou tell from the graph of a quadratic equation whether or not it has real solutions or imaginar solutions? Remember, the set of comple numbers includes both real and imaginar numbers. So, some solutions are real, some are imaginar, but all solutions are comple. 15. Do ou think ou can determine the imaginar solutions b eamining the graph? Eplain our reasoning. 3. Recall the function p() 5 1. a. Use an method to solve Let's see... I know how to analze a graph, complete the square, factor, and use the Quadratic Formula. Which method works best here? 15.5 Solving Quadratics with Comple Solutions 1117
6 15 b. Consider a function written in the form a 1 c 5 0. Complete the table to show when the solutions of a function are real or imaginar. c is positive c is negative a is positive a is negative. Recall the function q() 5 1. a. Use an method to solve What do the variables a and c tell ou about the graphical behavior of the function? Will the quadratic pass through the -ais? b. Suppose ou use the Quadratic Formula to solve the equation in part (a). How can ou tell whether the solutions are real or imaginar? Here's a hint: Look at the discriminant. 111 Chapter 15 Real Number Sstems
7 Problem 3 Imaginar fi Impossible Consider the function f() In what form is the quadratic function given?. Determine the -intercept of the function. 3. Use an method to determine the zeros of the function.. Are the zeros of the function real or imaginar? Eplain how ou know Solving Quadratics with Comple Solutions 1119
8 15 Recall that a quadratic function in factored form is written in the form f() 5 a( r 1 )( r ). 5. What do r 1 and r represent for a function written in this form?. Use our answer to Question 3 to write the function f() 5 1 in factored form. 7. Is the function ou wrote in factored form the same as the original function in standard form? Simplif the function ou wrote in Question to verif our answer. Can I still use the Distributive Propert with imaginar numbers? 110 Chapter 15 Real Number Sstems
9 Recall that the ais of smmetr is the vertical line that passes through the verte of a parabola and divides it in half.. Eplain how to determine the ais of smmetr using the zeros of the function Determine the ais of smmetr for f(). Show our work. 10. Use the ais of smmetr to determine the verte of f(). Show our work. Remember that the verte is located on the ais of smmetr. Recall that a quadratic function in verte form is written in the form f() 5 a( h) 1 k with verte (h, k). 11. Rewrite the function f() 5 1 in verte form. 1. Is the function ou wrote in verte form the same as the original function in standard form? Simplif the function ou wrote in Question 11 to check. He! Everthing we have learned about quadratic functions is still true even if the solutions are imaginar! 15.5 Solving Quadratics with Comple Solutions 111
10 15 Talk the Talk?1. Case sas that an quadratic equation has onl one of these 3 tpes of solutions: unique real number solutions equal real number solutions (a double root) 1 real and 1 imaginar solution Brandon sas that an quadratic equation has onl one of these 3 tpes of solutions: unique real number solutions equal real number solutions (a double root) imaginar solutions Karl sas that an quadratic equation has onl one of these tpes of solutions: unique real number solutions equal real number solutions (a double root) imaginar solutions 1 real and 1 imaginar solution Who s correct? Eplain our reasoning.. Eplain wh it is not possible for a quadratic equation to have equal imaginar solutions (double imaginar root). Be prepared to share our solutions and methods. 11 Chapter 15 Real Number Sstems
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