# Lesson 24: Using the Quadratic Formula,

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1 , b ± b 4ac x = a Opening Exercise 1. Examine the two equation below and discuss what is the most efficient way to solve each one. A. 4xx + 5xx + 3 = xx 3xx B. cc 14 = 5cc. Solve each equation with the most efficient method. 3. What are the differences between these two quadratic equations? Is one easier to solve than the other? Explain your thinking. 4. Is one pathway to the solution more correct than another? Unit 1: Completing the Square and The Quadratic Formula S.175

2 Exploration - Different Types of Solutions 5. A. Solve xx 1 = 0. B. What method is the most efficient for solving this problem? Why? 6. A. Solve xx xx + = B. What problem did you run into with this equation? 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? Unit 1: Completing the Square and The Quadratic Formula S.176

3 7. A. Solve xx xx + 1 = B. At the right is a graph of the equation xx xx + 1 = y. How does the graph support your findings in Part A? Describe the solutions of these quadratic equations in your own words. What makes these equations different? In the Quadratic Formula, x b ± b 4ac a =, the expression under the radical is called the discriminant: bb 4aacc. The value of the discriminant determines the number and nature of the solutions for a quadratic equation. 9. Determine the discriminant for the equations in Exercises 5, 6 and 7. 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 Number of solutions (x-intercepts) Unit 1: Completing the Square and The Quadratic Formula S.177

4 10. 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? 11. Fill in the blanks for the rules for discriminants. A. As we saw in the Exercise 5, 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 6, then we have ± (negative number), which can never lead to a solution. C. When the discriminant equals zero, as it did in Exercise 7, then we have ± 0, which yields only solution, bb aa. Unit 1: Completing the Square and The Quadratic Formula S.178

5 Practice Exercises For Exercises 1 15, determine the number of real solutions for each quadratic equation without solving. 1. pp + 7pp + 33 = 8 3pp 13. 7xx + xx + 5 = yy + 10yy = yy + 4yy zz + 9 = 4zz 16. 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 Discriminant A: ( ) 4(1)() Discriminant B: ( 4) 4( 1)( 4) Discriminant C: ( 4) 4(1)(0) Discriminant D: ( 8) 4( 1)( 13) Graph: Graph: Graph: Graph: Unit 1: Completing the Square and The Quadratic Formula S.179

6 17. 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. 18. Extension: Consider the quadratic equation aaxx + bbxx + cc = 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 1: Completing the Square and The Quadratic Formula S.180

7 Lesson Summary You can use the sign of the discriminant, bb 4aacc, to determine the number of real solutions to a quadratic equation in the form aaxx + bbxx + cc = 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 xx 3xx = 5 + xx xx 4. 4qq + 7 = qq 5qq + 1 Unit 1: Completing the Square and The Quadratic Formula S.181

8 Based on the graph of each quadratic function, yy = ff(xx), determine the number of real solutions for each corresponding quadratic equation, ff(xx) = Unit 1: Completing the Square and The Quadratic Formula S.18

9 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 c. Show that the function from part (b) written in factored form is equivalent to the original function d. Graph the equation Unit 1: Completing the Square and The Quadratic Formula S.183

10 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 d. Graph the equation Unit 1: Completing the Square and The Quadratic Formula S.184

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