, b ± b 4ac x = a Opening Exercise Over the years, many students and teachers have thought of ways to help us all remember the quadratic formula. Below is the YouTube link to a video created by two teachers at Sequoyah High School near Canton, Georgia. They used Adelle s Rolling in the Deep with modified lyrics, as shown at the right. https://www.youtube.com/watch?v=z6hcu0eps-o The a, b and c in the quadratic formula refers to the coefficients when the quadratic equation is in the form, 0 = ax + bx + c. 1. Read through the lyrics and highlight where the quadratic formula is.. According to their song, what does the discriminant tell you? 3. What do you have to make sure you have before you can use the quadratic formula? Unit 10: Completing the Square and The Quadratic Formula S.41
In Lesson 4 we reexamined two ways to find x-intercepts factoring and completing the square. In this lesson, we ll use the quadratic formula, b ± b 4ac x =, to find the x-intercepts. In Lesson 6, we ll see a how the quadratic formula came about. Let s practice using the quadratic formula. Follow the steps below to solve for x. 4. x = x + 3 Steps Work A. Write the equation in standard form, ax + bx + c = 0 B. Identify the a, b and c. a =, b =, c= C. Substitute the values for a, b and c into the quadratic equation. ± ( ) 4 g g x = g D. Simplify. E. Can you simplify the radical? F. State both answers. x = ; x = Unit 10: Completing the Square and The Quadratic Formula S.4
Practice Problems Solve each quadratic equation. Some will have radical answers. 5. 6x = 7x + 3 6. x = x 7. 5x + 6x = 1 8. x = x Discussion 9. Which of the quadratic equations in Exercises 5 8 could be factored to find the solutions? Which method do you think would be easier (factoring or using the quadratic formula)? Unit 10: Completing the Square and The Quadratic Formula S.43
Exploration - Different Types of Solutions 10. A. Solve xx 1 = 0. B. What method is the most efficient for solving this problem? Why? 11. A. Solve xx xx + = 0. 10 9 8 B. What problem did you run into with this equation? 7 6 5 4 C. At the right is a graph of the equation xx xx + = y. How does the graph support your findings in Parts A and B? 3 1 0-3 - -1 0 1 3 4 5 Unit 10: Completing the Square and The Quadratic Formula S.44
1. A. Solve xx xx + 1 = 0. 10 9 8 7 6 5 B. At the right is a graph of the equation xx xx + 1 = y. How does the graph support your findings in Part A? 4 3 1 0-3 - -1 0 1 3 4 5 13. Describe the solutions of these quadratic equations in your own words. What makes these equations different? In the Quadratic Formula, x b ± b 4ac =, the expression under the radical is a called the discriminant: bb 4aaaa. The value of the discriminant determines the number and nature of the solutions for a quadratic equation. 14. Determine the discriminant for the equations in Exercises 10, 11 and 1. What pattern do you notice? xx 1 = y xx xx + = y xx xx + 1 = y Values of a, b and c a = ; b = ; c = a = ; b = ; c = a = ; b = ; c = Discriminant: b 4ac Unit 10: Completing the Square and The Quadratic Formula S.45
Number of solutions (x-intercepts) 15. What are the differences among these three graphs? Which of these graphs belongs to a quadratic equation with a positive discriminant? Which belongs to a quadratic equation with a negative discriminant? Which graph has a discriminant equal to zero? 16. Fill in the blanks for the rules for discriminants. A. As we saw in the Exercise 10, when the discriminant is positive, then we have ± (positive number), which yields real solutions. If the discriminant is a square number, then we get two solutions. B. When the discriminant is a negative number, as in Exercise 11, then we have ± (negative number), which gives us a solution. C. When the discriminant equals zero, as it did in Exercise 1, then we have ± 0, which yields only solution, bb aa. Unit 10: Completing the Square and The Quadratic Formula S.46
Practice Exercises For Exercises 17 0, determine the number of real solutions for each quadratic equation without solving. 17. pp + 7pp + 33 = 8 3pp 18. 7xx + xx + 5 = 0 19. yy + 10yy = yy + 4yy 3 0. 4zz + 9 = 4zz 1. On the line below each graph, state whether the discriminant of each quadratic equation is positive, negative, or equal to zero. Then, identify which graph matches the discriminants below. Graph 1 Graph Graph 3 Graph 4 Unit 10: Completing the Square and The Quadratic Formula S.47
Discriminant A: Discriminant B: Discriminant C: Discriminant D: ( ) 4(1)() ( 4) 4( 1)( 4) ( 4) 4(1)(0) ( 8) 4( 1)( 13) Graph: Graph: Graph: Graph:. Consider the quadratic function ff(xx) = xx xx 4. A. Use the quadratic formula to find the xx-intercepts of the graph of the function. B. Use the xx-intercepts to write the quadratic function in factored form. C. Show that the function from Part B written in factored form is equivalent to the original function. 3. CHALLENGE: Consider the quadratic equation aaxx + bbxx + aa = 0. A. Write the equation in factored form, aa(xx mm)(xx nn) = 0, where mm and nn are the solutions to the equation. B. Show that the equation from Part A is equivalent to the original equation. Unit 10: Completing the Square and The Quadratic Formula S.48
Lesson Summary The quadratic formula, xx = bb ± bb 4aaaa, is one way to solve a quadratic equation of the form aa aaxx + bbxx + aa = 0, where aa 0. The formula can be used to solve any quadratic equation, and is especially useful for those that are not easily solved using any other method (i.e., by factoring or completing the square). You can use the sign of the discriminant, bb 4aaaa, to determine the number of real solutions to a quadratic equation in the form aaxx + bbxx + aa = 0, where aa 0. If the equation has a positive discriminant, there are two real solutions. A negative discriminant yields no real solutions. A discriminant equal to zero yields only one real solution. Homework Problem Set Without solving, determine the number of real solutions for each quadratic equation. 1. bb 4bb + 3 = 0. nn + 7 = 4nn + 5 3. xx 3xx = 5 + xx xx 4. 4qq + 7 = qq 5qq + 1 Unit 10: Completing the Square and The Quadratic Formula S.49
Based on the graph of each quadratic function, yy = ff(xx), determine the number of real solutions for each corresponding quadratic equation, ff(xx) = 0. 5. 6. 7. 8. Unit 10: Completing the Square and The Quadratic Formula S.50
9. Consider the quadratic function ff(xx) = xx 7. A. Find the xx-intercepts of the graph of the function. B. Use the xx-intercepts to write the quadratic function in factored form. 10 9 C. Show that the function from part (b) written in factored form is equivalent to the original function. 8 7 6 5 4 3 1 0-5 -4-3 - -1-1 0 1 3 4 5 - -3-4 -5-6 d. Graph the equation. -7-8 -9-10 Unit 10: Completing the Square and The Quadratic Formula S.51
10. Consider the quadratic function ff(xx) = xx + xx + 5. A. Find the xx-intercepts of the graph of the function. B. Use the xx-intercepts to write the quadratic function in factored form. 10 C. Show that the function from part (b) written in factored form is equivalent to the original function. 9 8 7 6 5 4 3 1 0-5 -4-3 - -1-1 0 1 3 4 5 - -3-4 -5 D. Graph the equation. -6-7 -8-9 -10 Unit 10: Completing the Square and The Quadratic Formula S.5
11. Examine the two equation below and decide what is the most efficient way to solve each one. A. 4xx + 5xx + 3 = xx 3xx B. xx 14 = 5xx 1. Solve each equation with the most efficient method. 13. What are the differences between these two quadratic equations? Is one easier to solve than the other? Explain your thinking. 14. Is one pathway to the solution more correct than another? Unit 10: Completing the Square and The Quadratic Formula S.53
Unit 10: Completing the Square and The Quadratic Formula S.54