11.3 Finding Complex Solutions of Quadratic Equations

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1 Name Class Date 11.3 Finding Complex Solutions of Quadratic Equations Essential Question: How can you find the complex solutions of any quadratic equation? Resource Locker Explore Investigating Real Solutions of Quadratic Equations A Complete the table. ax + bx + c = 0 ax + bx = -c f(x) = ax + bx g(x)= -c x + x + 1 = 0 x + x + = 0 x + x + 3 = 0 B The graph of ƒ(x)= x + x is shown. Graph each g(x). Complete the table. y x Equation x + x + 1 = 0 x + x + = 0 Number of Real Solutions x + x + 3 = 0 C Repeat Steps A and B when ƒ(x) = -x + x. - ax + bx + c = 0 ax + bx = -c f(x) = ax + bx g(x)= -c -x + x - 1 = 0 -x + x - = 0 -x + x - 3 = y x Equation -x + x - 1 = 0 -x + x - = 0 -x + x - 3 = 0 Number of Real Solutions Module Lesson 3

2 Reflect 1. Look back at Steps A and B. Notice that the minimum value of f(x) in Steps A and B is -. Complete the table by identifying how many real solutions the equation ƒ(x) = g(x)has for the given values of g(x). Value of g(x) g(x) = - g(x) > - g(x) < - Number of Real Solutions of f(x)= g(x). Look back at Step C. Notice that the maximum value of ƒ(x)in Step C is. Complete the table by identifying how many real solutions the equation ƒ (x) = g(x) has for the given values of g(x). Value of g(x) g(x) = g(x) > g(x) < Number of Real Solutions of f(x)= g(x) 3. You can generalize Reflect 1: For ƒ(x) = ax + bx where a > 0, ƒ(x) = g(x) where g(x) = -c has real solutions when g(x)is greater than or equal to the minimum value of ƒ(x). The minimum value of ƒ(x) is ƒ( b_ - a ) = a ( - b_ a) + b( - b_ a) ( = a _ b a ) - _ b a = _ b a - _ b a = _ b a - _ b a = - _ b a. So, ƒ(x) = g(x) has real solutions when g(x) - _ b a. Substitute -c for g(x). -c - _ b a Add b a to both sides. _ b a - c 0 Multiply both sides by a, which is positive. b - ac 0 In other words, the equation ax + bx + c = 0 where a > 0 has real solutions when b - ac 0. Generalize the results of Reflect in a similar way. What do you notice? Module Lesson 3

3 Explain 1 Finding Complex Solutions by Completing the Square Recall that completing the square for the expression x + bx requires adding ( to it, resulting in the perfect square ) ) trinomial x + bx + (, which you can factor as (x +. Don t forget that when x + bx appears on one side of an equation, adding ( ) Example 1 ) to it requires adding ( to the other side as well. Solve the equation by completing the square. State whether the solutions are real or non-real. ) A 3 x + 9x - 6 = 0 1. Write the equation in the form x + bx = c. 3 x + 9x - 6 = 0 3 x + 9x = 6 x + 3x =. Identify b and ( ). ( b_ ) ) = ( 3_ ) b = 3 = 9_ 3. Add ( to both sides of the equation. x + 3x + 9_ = + 9_. Solve for x. ( x + 3_ = + _ ) 9 ( x + 3_ ) = _ 17 = ± _ 17 x + 3 x + 3_ = ± _ 17 x = - 3_ ± 17 x = -3 ± 17 There are two real solutions: and. B x - x + 7 = 0 1. Write the equation in the form x + bx = c.. Solve for x.. Identify b and ( ). b = b_ ( = ) ( ) _ = 3. Add ( ) to both sides. x - x + = -7 + x + x = -7 + (x- ) = x - = ± x = 1 ± There are two real/non-real solutions: and. Module Lesson 3

4 Reflect. How many complex solutions do the equations in Parts A and B have? Explain. Your Turn Solve the equation by completing the square. State whether the solutions are real or non-real. 5. x + 8x + 17 = 0 6. x + 10x - 7 = 0 Explain Identifying Whether Solutions Are Real or Non-real By completing the square for the general quadratic equation ax + bx + c = 0, you can obtain the quadratic b - ac formula, x =, which gives the solutions of the general quadratic equation. In the quadratic formula, the a expression under the radical sign, b - ac, is called the discriminant, and its value determines whether the solutions of the quadratic equation are real or non-real. -b ± Value of Discriminant b - ac > 0 b - ac = 0 b - ac < 0 Example A Number and Type of Solutions Two real solutions One real solution Two non-real solutions Answer the question by writing an equation and determining whether the solutions of the equation are real or non-real. A ball is thrown in the air with an initial vertical velocity of 1 m/s from an initial height of m. The ball s height h (in meters) at time t (in seconds) can be modeled by the quadratic function h (t) = -.9 t + 1t +. Does the ball reach a height of 1 m? Set h (t) equal to t + 1t + = 1 Subtract 1 from both sides. -.9 t + 1t + 10 = 0 Image Credits: Fred Fokkelman/Shutterstock Find the value of the discriminant. 1 - (-.9) (-10) = = 0 Because the discriminant is zero, the equation has one real solution, so the ball does reach a height of 1 m. Module Lesson 3

5 B A person wants to create a vegetable garden and keep the rabbits out by enclosing it with 100 feet of fencing. The area of the garden is given by the function A (w) = w (50 - w) where w is the width (in feet) of the garden. Can the garden have an area of 700 ft? Set A (w) equal to 700. w (50 - w) = Multiply on the left side. 50w - w = Subtract 700 from both sides. - w + 50w - = 0 Find the value of the discriminant. Because the discriminant is [positive/zero/negative], the equation has [two real/one real/two non-real] solutions, so the garden [can/cannot] have an area of 700 ft. Your Turn Answer the question by writing an equation and determining if the solutions are real or non-real. 7. A hobbyist is making a toy sailboat. For the triangular sail, she wants the height h (in inches) to be twice the length of the base b (in inches). Can the area of the sail be 10 i n? Image Credits: David Burton/Alamy Explain 3 Finding Complex Solutions Using the Quadratic Formula When using the quadratic formula to solve a quadratic equation, be sure the equation is in the form ax + bx + c = 0. Example 3 Solve the equation using the quadratic formula. Check a solution by substitution. A -5 x - x - 8 = 0 -b ± Write the quadratic formula. x = b - ac a Substitute values. = -(-) ± (-) - (-5)(-8) (-5) ± ± i 39 Simplify. = = _ Module Lesson 3

6 So, the two solutions are _ i _ 39 5 and _ i _ Check by substituting one of the values. Substitute. -5(- 5 - i _ 39 ) - ( i _ 39 ) Square. -5( 5 + i _ 39 -_ ) - (- 5 - i _ 39 ) Distribute i _ 39 + _ _ 5 + i _ Simplify. 0 _ = 0 B 7x + x + 3 = -1 Write the equation with 0 on one side. 7x + x + = 0 b Write the quadratic formula. x = - ac a - ± ( ) - ( )( ) Substitute values. = ( ) -b ± - Simplify. = 1 = - ± - ± i - ± i = 1 7 So, the two solutions are and. Check by substituting one of the values. Substitute. Square. Distribute. Simplify. Module Lesson 3

7 Your Turn Solve the equation using the quadratic formula. Check a solution by substitution. 8. 6x - 5x - = 0 9. x + 8x + 1 = x Elaborate 10. Discussion Suppose that the quadratic equation ax + bx + c = 0 has p + qi where q 0 as one of its solutions. What must the other solution be? How do you know? 11. Discussion You know that the graph of the quadratic function ƒ(x) = ax + bx + c has the vertical line b x = - as its axis of symmetry. If the graph of ƒ(x) crosses the x-axis, where do the x-intercepts occur a relative to the axis of symmetry? Explain. 1. Essential Question Check-In Why is using the quadratic formula to solve a quadratic equation easier than completing the square? Module Lesson 3

8 Evaluate: Homework and Practice 1. The graph of ƒ (x) = x + 6x is shown. Use the graph to determine how many real solutions the following equations have: x + 6x + 6 = 0, x + 6x + 9 = 0, and x + 6x + 1 = 0. Explain y x Online Homework Hints and Help Extra Practice -1. The graph of ƒ (x) = - x + 3x is shown. Use the graph to determine how many real solutions the following equations have: - x + 3x - 3 = 0, - x + 3x - 9_ = 0, and - x + 3x - 6 = 0. Explain y x Solve the equation by completing the square. State whether the solutions are real or non-real. 3. x + x + 1 = 0. x + x + 8 = 0 Module Lesson 3

9 5. x - 5x = x - 6x = x + 13x = x - 6x - 11 = 0 Without solving the equation, state the number of solutions and whether they are real or non-real x + x + 13 = x - 11x + 10 = x - _ 5 x = 1 1. x + 9 = 1x Module Lesson 3

10 Answer the question by writing an equation and determining whether the solutions of the equation are real or non-real. 13. A gardener has 10 feet of fencing to put around a rectangular vegetable garden. The function A (w) = 70w - w gives the garden s area A (in square feet) for any width w (in feet). Does the gardener have enough fencing for the area of the garden to be 1300 ft? 1. A golf ball is hit with an initial vertical velocity of 6 ft/s. The function h (t) = -16t + 6t models the height h (in feet) of the golf ball at time t (in seconds). Does the golf ball reach a height of 60 ft? 15. As a decoration for a school dance, the student council creates a parabolic arch with balloons attached to it for students to walk through as they enter the dance. The shape of the arch is modeled by the equation y = x (5 - x), where x and y are measured in feet and where the origin is at one end of the arch. Can a student who is 6 feet 6 inches tall walk through the arch without ducking? Image Credits: Aflo Foto Agency/Alamy Module Lesson 3

11 16. A small theater company currently has 00 subscribers who each pay $10 for a season ticket. The revenue from season-ticket subscriptions is $,000. Market research indicates that for each $10 increase in the cost of a season ticket, the theater company will lose 10 subscribers. A model for the projected revenue R (in dollars) from season-ticket subscriptions is R(p) = ( p)(00-10p), where p is the number of $10 price increases. According to this model, is it possible for the theater company to generate $5,600 in revenue by increasing the price of a season ticket? Solve the equation using the quadratic formula. Check a solution by substitution. 17. x - 8x + 7 = x - 30x+ 50 = 0 Module Lesson 3

12 19. x + 3 = x 0. x + 7 = x 1. Place an X in the appropriate column of the table to classify each equation by the number and type of its solutions. Equation Two Real Solutions One Real Solution Two Non-Real Solutions x - 3x + 1 = 0 x - x + 1 = 0 x - x +1 = 0 x + 1 = 0 x + x + 1 = 0 x + x + 1 = 0 x + 3x + 1 = 0 Module Lesson 3

13 H.O.T. Focus on Higher Order Thinking. Explain the Error A student used the method of completing the square to solve the equation -x + x - 3 = 0. Describe and correct the error. -x + x - 3 = 0 -x + x = 3 -x + x + 1 = (x + 1) = x + 1 = ± _ x + 1 = ± x = -1 ± So, the two solutions are -1 + = 1 and -1 - = Make a Conjecture Describe the values of c for which the equation x + 8x + c = 0 has two real solutions, one real solution, and two non-real solutions.. Analyze Relationships When you rewrite y = ax + bx + c in vertex form by b completing the square, you obtain these coordinates for the vertex: (- a, c - b a ). Suppose the vertex of the graph of y = ax + bx + c is located on the x-axis. Explain how the coordinates of the vertex and the quadratic formula are in agreement in this situation. Module Lesson 3

14 Lesson Performance Task Matt and his friends are enjoying an afternoon at a baseball game. A batter hits a towering home run, and Matt shouts, Wow, that must have been 110 feet high! The ball was feet off the ground when the batter hit it, and the ball came off the bat traveling vertically at 80 feet per second. a. Model the ball s height h (in feet) at time t (in seconds) using the projectile motion model h(t)= -16t + v 0 t + h 0 where v 0 is the projectile s initial vertical velocity (in feet per second) and h 0 is the projectile s initial height (in feet). Use the model to write an equation based on Matt s claim, and then determine whether Matt s claim is correct. b. Did the ball reach a height of 100 feet? Explain. c. Let h max be the ball s maximum height. By setting the projectile motion model equal to h max, show how you can find h max using the discriminant of the quadratic formula. d. Find the time at which the ball reached its maximum height. Module Lesson 3