Skill: solve quadratic equations by factoring. Solve Quadratic Equations A.SSE.A. Interpret the structure of epressions. Use the structure of an epression to identify ways to rewrite it. For eample, see 4 y 4 as ( ) (y ), thus recognizing it as a difference of squares that can be factored as ( y )( + y ). A.SSE.B.3a Write epressions in equivalent forms to solve problems. Choose and produce an equivalent form of an epression to reveal and eplain properties of the quantity represented by the epression. Factor a quadratic epression to reveal the zeros of the function it defines. A.REI.B.4b Solve equations and inequalities in one variable. Solve quadratic equations by inspection (e.g., for = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives comple solutions and write them as a ± bi for real numbers a and b. Review: Factoring Quadratic Trinomials into Two Binomials (Using the ac method or splitting the middle term.) NOTE: You may want to see Factor by Splitting the Middle Term notes AND Factoring Quadratic Trinomial notes on www.rpdp.net. Factoring a b c a, 1 E 0: Factor 7 1. Find two integers such that their product is 1 and their sum is 7. 4 and 3 Write the two binomials as a product. 4 3 Factoring a b c a, 1 E 1: Factor 7 3. Step One: Multiply a c. 3 6 Step Two: Find two integers such that their product is ac6 and their sum is b 7. 6 and 1 Step Three: Rewrite ( split ) the middle term as a sum of two terms using the numbers from Step Two. 6 1 3 Step Four: Factor by grouping. Group the first terms and last terms and factor out the GCF from each pair. SolvingQuadratics Page 1 of 17 3/16/015
6 1 3 3 1 3 Step Five: If Step Four was done correctly, there should be a common binomial factor. Factor this binomial out and write what remains from each term as the second binomial factor. 1 3 E : Factor 5 7. Step One: Multiply a c. 5 10 Step Two: Find two integers such that their product is ac10 and their sum is b 7. and 5 Step Three: Rewrite ( split ) the middle term as a sum of two terms using the numbers from Step Two. 5 5 Step Four: Factor by grouping. Group the first terms and last terms and factor out the GCF from each pair. 5 5 5 15 Step Five: If Step Four was done correctly, there should be a common binomial factor. Factor this binomial out and write what remains from each term as the second binomial factor. 5 1 Recall: Special Factoring Patterns Difference of Two Squares: a b a ba b Perfect Square Trinomial: a ab b a b a ab b a b SolvingQuadratics Page of 17 3/16/015
Zero Product Property: If the product of two factors is 0, then one or both of the factors must equal 0. E 3: Solve the equation 3 1 0 using the zero product property. Since one or both of the factors must equal 0, we will solve the two equations 3 0 and 1 0. 30 3 10 1 Solutions: 1, 3 Solving a Quadratic Equation by Factoring E 4: Solve the equation 5 6 by factoring. Step One: Write the equation in standard form. 5 6 0 Step Two: Factor the quadratic. 3 0 Step Three: Set each factor equal to zero and solve. Note: Check this answer by graphing on the calculator. 3 0 0 3, 3 E 5: Solve the equation 4 8. Step One: Write the equation in standard form. Step Two: Factor the quadratic using the ac method. 3 58 0 a c 4 b 5 8 and 3 3 8 3 8 0 3 8 1 0 3 8 1 3 8 0 Step Three: Set each factor equal to zero and solve. The solutions can be written in set notation: 3 8 0 1 0 8 3 1 8 1, 3 SolvingQuadratics Page 3 of 17 3/16/015
E 6: Solve the equation 5 30 9 y y. Step One: Write the equation in standard form. 9y 30y 5 0 Step Two: Factor the quadratic. 3y 30y5 y Note: 5 3 30y 3y 5 0 Step Three: Set each factor equal to zero and solve. The solution can be written in set notation: 3y 5 0 y 5 3 5 3 Zero(s) of Quadratic Functions: the -value(s) where the function intersects the -ais To find the zero(s), factor the quadratic and set each factor equal to 0. Note: We can graph quadratic functions by plotting the zeros. The verte is halfway between the zeros. E 7: Find the zero(s) of the quadratic function y 3and graph the parabola. Step One: Factor the quadratic polynomial. Step Two: Set each factor equal to 0 and solve. Step Three: Find the coordinates of the verte. y 3 3 1 y 3 0 1 0 3 1 3 1 1 y 1 1 3 1 3 4 Step Four: Plot the points and sketch the parabola. SolvingQuadratics Page 4 of 17 3/16/015
You Try: Solve the quadratic equation 5t 5 4t 6 by factoring. QOD: What must be true about a quadratic equation before you can solve it using the zero product property? Sample Practice Question(s): 1. What is the solution set for the following equation? 10 9 0 A. { 9, 1} B. { 9, 1} C. { 1, 9} D. {1, 9}. Which of the following equations has roots of 7 and 4? A. 7 4 0 B. 7 4 0 C. 7 4 0 D. 7 4 0 SolvingQuadratics Page 5 of 17 3/16/015
Skill: solve quadratic equations by completing the square. A.SSE.A. Interpret the structure of epressions. Use the structure of an epression to identify ways to rewrite it. A.SSE.B.3b Write epressions in equivalent forms to solve problems. Complete the square in a quadratic epression to reveal the maimum or minimum value of the function it defines. A.REI.B.4a Solve equations and inequalities in one variable. Solve quadratic equations in one variable. Use the method of completing the square to transform any quadratic equation in into an equation of the form ( p) = q that has the same solutions. Derive the quadratic formula from this form. Review: Factoring a Perfect Square Trinomial a ab b a b a ab b a b NOTE: You may want to review Completing the Square Notes on www.rpdp.net. Completing the Square: writing an epression of the form to factor it as a binomial squared To complete the square of b, we must add b as a perfect square trinomial in order b. Teacher Note: Algebra Tiles work well to illustrate completing the square. See Page 79 for an activity. E 8: Find the value of c such that 10 c is a perfect square trinomial. b 10, therefore we must add 10 5 5 Note: 10 5 c to complete the square. Solving a Quadratic Equation by Completing the Square E 9: Solve 8 7 0 by completing the square. Step One: Rewrite to make the lead coefficient 1. Step Two: Take the constant term to the other side. b Step Three: Complete the square (add to both sides). SolvingQuadratics Page 6 of 17 3/16/015 8 7 0 8 7 8 8 8 7 816 9 Step Four: Factor the perfect square trinomial. 4 9
Step Five: Take the square roots of both sides. 4 3 4 9 4 3 4 3 Step Si: Solve for the variable. 1 7 The solution set is 7, 1. Check your answer by factoring. E 30: Solve 1 4 0 by completing the square. Step One: Rewrite to make the lead coefficient 1. Step Two: Take the constant term to the other side. b Step Three: Complete the square (add to both sides). 1 4 0 6 0 6 6 6 6 6 9 11 Step Four: Factor the perfect square trinomial. 3 11 Step Five: Take the square roots of both sides. Step Si: Solve for the variable. The solution set is 3 11, 3 11 3 11 3 11 3 11 3 11 3 11 3 11 E 31: Solve 3 0 by completing the square. Step One: Rewrite to make the lead coefficient 1. Step Two: Take the constant term to the other side. b Step Three: Complete the square (add Step Four: Factor the perfect square trinomial. to both sides). 3 0 3 1 1 3 1 13 4 4 1 13 4 SolvingQuadratics Page 7 of 17 3/16/015
Step Five: Take the square roots of both sides. Step Si: Solve for the variable. 1 13 The solutions are. 1 13 4 1 13 1 13 1 13 1 13 1 13 You Try: Solve by completing the square. 4 011 0 QOD: Describe why adding b to b makes it a perfect square trinomial. Sample CCSD Common Eam Practice Question(s): What are the roots (solutions) of A. 1 3, 1 3 A. 1 5, 1 5 1 0? B. C. 1 5 1 5, 1 5 1 5, SolvingQuadratics Page 8 of 17 3/16/015
Skill: solve quadratic equations using the quadratic formula. A.SSEA.. Interpret the structure of epressions. Use the structure of an epression to identify ways to rewrite it. A.REI.B.4a Solve equations and inequalities in one variable. Use the method of completing the square to transform any quadratic equation in into an equation of the form ( p) = q that has the same solutions. Derive the quadratic formula from this form. A.REI.B.4b Solve equations and inequalities in one variable. Solve quadratic equations by inspection (e.g., for = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives comple solutions and write them as a ± bi for real numbers a and b. A.REI.A.1 Understand solving equations as a process of reasoning and eplain the reasoning. Eplain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Notes: Deriving the Quadratic Formula by Completing the Square Solve the quadratic equation a b c 0 by completing the square. Step One: Rewrite so that the lead coefficient is 1. Step Two: Take the constant term to the other side. b Step Three: Complete the square (add Step Four: Factor the perfect square trinomial. Step Five: Take the square roots of both sides. to both sides). a b c 0 a a a a b c 0 a a b c a a b b c b a a a a b b 4ac b a 4a 4a b b 4ac a 4a b b 4ac a 4a b b 4ac a 4a SolvingQuadratics Page 9 of 17 3/16/015
Step Si: Solve for the variable. b b 4ac b b 4ac a a a a a a b b 4ac b b 4ac The Quadratic Formula: To solve a quadratic equation in the form b b 4ac. a a b c 0, use the formula Note: To help memorize the quadratic formula, sing it to the tune of the song Pop Goes the Weasel. E 3: Solve the quadratic equation 8 1 using the quadratic formula. Step One: Rewrite in standard form (if necessary). 81 0 Step Two: Identify a, b, and c. a 1, b 8, c 1 Step Three: Substitute the values into the quadratic formula. b b 4ac a 8 8 411 1 8 64 4 8 60 8 15 Step Four: Simplify. 4 15 The solution set is 4 15,4 15 E 33: Solve the quadratic equation 5 1 6 using the quadratic formula. Step One: Rewrite in standard form (if necessary). 6 51 0 Step Two: Identify a, b, and c. a 6, b 5, c 1 Step Three: Substitute the values into the quadratic formula. b b 4ac a 5 5 46 1 6 SolvingQuadratics Page 10 of 17 3/16/015
Step Four: Simplify. The solution set is 1 1, 6 5 5 4 5 49 5 7 1 1 1 5 7 1 5 7 1 1 1 1 1 1 6 E 34: Solve the quadratic equation 1 4 9 0 using the quadratic formula. Step One: Rewrite in standard form (if necessary). 9 1 4 0 Step Two: Identify a, b, and c. a 9, b 1, c 4 Step Three: Substitute the values into the quadratic formula. Step Four: Simplify. The solution set is 3. b b 4ac a 1 1 494 9 1 144 144 1 0 1 18 18 18 3 E 35: Solve the quadratic equation 3 0 using the quadratic formula. Step One: Rewrite in standard form (if necessary). 3 0 Step Two: Identify a, b, and c. a, b, c 3 Step Three: Substitute the values into the quadratic formula. b b 4ac a 43 4 4 0 Step Four: Simplify. 4 4 There is no real solution to the quadratic equation because 0 is not a real number. SolvingQuadratics Page 11 of 17 3/16/015
You Try: Solve the equation 6 3 using the quadratic formula. QOD: Write a conjecture about how the radicand in the quadratic formula relates to the number of solutions that a quadratic equation has. Sample Practice Question(s): 1. What are the roots (solutions) of A. 1 3, 1 3 B. 1 5, 1 5 1 0? C. D. 1 5 1 5, 1 5 1 5,. Which of the following is the correct use of the quadratic formula to find the solutions of the equation 7 5? A. B. C. D. 7 7 45 7 7 4 5, 7 7 45 7 7 4 5, 7 7 45 7 7 4 5, 7 7 45 7 7 4 5, SolvingQuadratics Page 1 of 17 3/16/015
Discriminant: the discriminant of the quadratic equation a b c 0 is b 4ac Note: The discriminant is the radicand of the quadratic formula! Determining the Number of Real Solutions of a Quadratic Equation Using the Discriminant Teacher Note: Students should have come up with this in the QOD. If If If b b b 4ac 0, then there are no real solutions. 4ac 0, then there is one solution. 4ac 0, then there are two real solutions. E 36: Determine the number of real solutions that the equations have. 1. 3 1 Rewrite the equation in standard form. 3 1 0 a 3, b 1, c 1 Find the discriminant. b ac 4 1 4 3 1 11 11 Determine the number of real solution(s). b 4ac 11 0, so there are no real solutions.. 45 Rewrite the equation in standard form. 5 4 0 a 1, b 5, c 4 Find the discriminant. b ac 4 5 4 1 4 5 16 41 Determine the number of real solution(s). b 4ac 41 0, so there are two real solutions. 3. 9 1 4 0 Rewrite the equation in standard form. 9 1 4 0 a 9, b 1, c 4 Find the discriminant. b ac 4 1 4 9 4 144 144 0 Determine the number of real solution(s). b 4ac 0, so there is one real solution. SolvingQuadratics Page 13 of 17 3/16/015
Determining the Number of -Intercepts of a Quadratic Function Using the Discriminant Because the -intercepts of y a b c are the same as the zeros of the equation a b c 0, we can use the discriminant to determine the number of -intercepts that a quadratic function has. E 37: Sketch the graph of a quadratic function with a negative discriminant. Because the discriminant, b 4ac 0, the function will have no -intercept. A sample answer is shown in the graph. Note: Any parabola which does not intersect the -ais is an acceptable answer. Application Problem E 38: A baton twirler tosses a baton into the air. The baton leaves the twirler s hand 6 feet above the ground and has an initial vertical velocity of 45 feet per second. This can be modeled by the equation h 16t 45t 6, where h is the height (in feet) and t is the time (in seconds). The twirler wants her baton to reach at least 40 feet. Will the baton reach that height? Substitute h 1. Write in standard form. 40 16t 45t 6 0 16t 45t 34 a 16, b 45, c 34 Find the discriminant. b ac 4 45 4 16 34 05 176 151 Since the discriminant is less than 0, this equation has no real solution. Therefore, the baton could not reach 40 feet. How high will the baton reach? Graph the function h t t 16 45 6. Find the maimum (verte). The baton will reach approimately 37.64 ft. You Try: Find values for c so that the equation will have no real solution, one real solution, and two real solutions. c 3 0 QOD: Write a quadratic equation which can be factored. Find its discriminant. Teacher Note: Have students share their answers to the QOD and allow students to make a conjecture for how to determine if a quadratic polynomial is factorable using the discriminant. (It must be a perfect square.) SolvingQuadratics Page 14 of 17 3/16/015
Sample Practice Question(s): The graph of y 1 has how many -intercepts? A. 1 B. C. 1 D. 0 Transformations of quadratic functions 1. How would the graph of the function be affected if the function were changed to? a. The graph would shift 5 units to the left. b. The graph would shift 5 units down. c. The graph would shift 5 units up. d. The graph would shift 3 units down.. How would the graph of the function be affected if the function were changed to? a. The graph would shift units up. b. The graph would shift 5 units up. c. The graph would shift units to the right. d. The graph would shift 5 unit down. 3. How would you translate the graph of to produce the graph of a. translate the graph of down 4 units b. translate the graph of up 4 units c. translate the graph of left 4 units d. translate the graph of right 4 units 4. Compare the graph of with the graph of. a. The graph of g() is wider. b. The graph of g() is narrower. c. The graph of g() is translated 6 units down from the graph of f(). d. The graph of g() is translated 6 units up from the graph of f(). 5. Compared to the graph of, the graph of is. a. narrower and translated down c. wider and translated down b. narrower and translated up d. wider and translated up SolvingQuadratics Page 15 of 17 3/16/015
6. Four bowls with the same height are constructed using quadratic equations as their shapes. Which bowl has the narrowest opening? a. c. Bowl Bowl b. Bowl d. Bowl 7. What is the parent function for? a. c. b. d. 8. Use the quadratic function f() = ( + 3)( 1), to graph the function. a. y c. 10 8 6 4 10 8 6 4 y 10 8 6 4 4 6 8 10 4 6 8 10 10 8 6 4 4 6 8 10 4 6 8 10 b. 10 y d. 10 y 8 8 6 6 4 4 10 8 6 4 4 6 8 10 4 6 8 10 10 8 6 4 4 6 8 10 4 6 8 10 9. Use the quadratic formula to solve the equation Describe how you can use the graph of the equation to verify your solutions of the equation Sketch this graph and verify your solutions. SolvingQuadratics Page 16 of 17 3/16/015
10. Eplain the zero-product property and why it is useful. Give an eample to illustrate your eplanation. Sample answer: The zero-product property states that if a product of factors is equal to zero, then one or more of the factors is equal to zero. For eample, in the equation the polynomial can be factored as ( + 7)( ). Since this factored polynomial is equal to zero, then the value of at least one of the factors is 0. This fact is useful because you can easily solve the equations and to find the solutions = 7 and =. Construct viable arguments for the following reflection questions. 1. Describe how to use the discriminant to find the number of real solutions to a quadratic equation.. Choose a method to solve 5 4 0 and eplain why you chose that method. 3. Describe how the discriminant can be used to determine if an object will reach a given height. 4. Eplain why there are no solutions to the quadratic equation 5. Describe how to estimate the solutions of 4 16 6. How do you find the zeros of a function from its graph? 9.. What are the approimate solutions? 7. Describe how to find the ais of symmetry of a quadratic function if its graph crosses the ais. Describe how to find the ais of symmetry of a quadratic function if its graph does not cross the ais? 8. Eplain how to graph a quadratic function. 9. What do the verte and zeros represent in the following situation. As Joe dives into his pool, his height in feet above the water can be modeled by the function f ( ) 16 16 1 where is the time in seconds after he begins diving. Find the maimum height of the dive and the time it takes Joe to reach this height. Then find how long it takes him to reach the water. 10. Eplain two ways to solve 6 0. How are these two methods similar? 11. Describe the relationships among the solutions, the zeros, and the -intercepts of y 4 1. SolvingQuadratics Page 17 of 17 3/16/015