# Lesson 23: Deriving the Quadratic Formula

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1 : Deriving the Quadratic Formula Opening Exercise 1. Solve for xx. xx 2 + 2xx = 8 7xx 2 12xx + 4 = 0 Discussion 2. Which of these problems makes more sense to solve by completing the square? Which makes more sense to solve by factoring? How could you tell early in the problem solving process which strategy to use? : Deriving the Quadratic Formula Unit 12: Completing the Square & The Quadratic Formula S.167 This file derived from ALG I--TE

3 aaxx 22 + bbxx + cc = 00 The steps: : Deriving the Quadratic Formula Unit 12: Completing the Square & The Quadratic Formula S.169 This file derived from ALG I--TE

4 Let s test the Quadratic Formula with the problems from the Opening Exercise. 7. Solve for xx, using the Quadratic Formula. Be sure to put each equation in standard form. xx = bb± bb2 4aaaa, for aaxx 2 + bbxx + cc = 0 2aa xx 2 + 2xx = 8 7xx 2 12xx + 4 = 0 a = ; b = ; c = a = ; b = ; c = 8. Did your answers in Exercise 7 agree with your answers in Exercise 1? 9. Let s look a little more closely at the Quadratic Formula. Notice that the whole expression can be split into two separate expressions as follows: bb ± bb 22 44aacc 22aa = bb 22aa ± bb22 44aacc. 22aa This part is the axis of! This part describes the location of the -! : Deriving the Quadratic Formula Unit 12: Completing the Square & The Quadratic Formula S.170 This file derived from ALG I--TE

5 Practice Exercises Use the quadratic formula to solve each equation. 10. xx 2 2xx = 12 aa = 1, bb = 2, cc = 12 (Watch the negatives.) rr2 6rr = 2 aa = 1, bb = 6, cc = 2 (Did you remember the negative?) pp 2 + 8pp = 7 aa = 2, bb = 8, cc = yy 2 + 3yy 5 = 4 aa = 2, bb = 3, cc = 9 : Deriving the Quadratic Formula Unit 12: Completing the Square & The Quadratic Formula S.171 This file derived from ALG I--TE

6 Solve these quadratic equations, using a different method for each: solve by factoring, solve by completing the square, and solve using the quadratic formula. Before starting, indicate which method you will use for each. 14. Method 15. Method 16. Method 22xx xx 33 = 00 xx xx 55 = xx22 xx 44 = 00 Lesson Summary The quadratic formula, xx = bb ± bb2 4aaaa, is derived by completing the square on the general 2aa form of a quadratic equation: aaxx 2 + bbxx + cc = 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). : Deriving the Quadratic Formula Unit 12: Completing the Square & The Quadratic Formula S.172 This file derived from ALG I--TE

7 Homework Problem Set Use the quadratic formula to solve each equation. Be sure to write each equation in standard form. 1. Solve for zz: zz 2 3zz 8 = Solve for qq: 2qq 2 8 = 3qq 3. Solve for mm: 1 3 mm2 + 2mm + 8 = Determine the error in Sergio s work below. 0 = 3x 2 4x ± (-4) -4(3)(-5) x = 2(3) -4± x = 6-4± 76-4±2 19-2± 19 x = = = x= or x= 3 3 : Deriving the Quadratic Formula Unit 12: Completing the Square & The Quadratic Formula S.173 This file derived from ALG I--TE

8 Solve these quadratic equations, using a different method for each: solve by factoring, solve by completing the square, and solve using the quadratic formula. Before starting, indicate which method you will use for each. 5. Method 6. Method 7. Method 3xx xx 10 = 0 xx 2 12xx + 28 = 0 2xx 2 9xx 9 = 0 8. Geometry Connection Determine the base and height of the right triangle below. Its area is 8 cm 2. [source: x + 2 2x : Deriving the Quadratic Formula Unit 12: Completing the Square & The Quadratic Formula S.174 This file derived from ALG I--TE

: Complicated Quadratics Opening Discussion 1. The quadratic expression 2x 2 + 4x + 3 can be modeled with the algebra tiles as shown below. Discuss with your group a method to complete the square with

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