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Objectives: The learner will…,

10.5 The Quadratic Formula. Objectives: The learner will…,. NCSCOS. Use the quadratic formula to find the zeros of a quadratic function. Evaluate the discriminate to determine how many real roots a quadratic equation has and whether it can be solved. 4.02. – b   b 2 – 4 ac.

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Objectives: The learner will…,

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  1. 10.5 The Quadratic Formula Objectives: The learner will…, NCSCOS • Use the quadratic formula to find the zeros of a quadratic function. • Evaluate the discriminate to determine how many real roots a quadratic equation has and whether it can be solved. • 4.02

  2. –b b2 – 4ac –b– b2 + (– 4ac) –b+ b2 + (– 4ac) x = x = x = 2a 2a 2a 10.5 The Quadratic Formula Quadratic Formula For ax2 + bx + c = 0, where a 0:

  3. –bb2 + (– 4ac) x = 2a x = –3  49 2 10.5 The Quadratic Formula Use the quadratic formula to solve: x2 + 3x – 10 = 0 a = 1, b = 3, c = –10 x = –(3) √(3)2+(–4)(1)(–10) 2(1) –3 – 7 –3 + 7 = = 2 2 4 –10 = = 2 2 –3 ± 7 x = 2 x = 2 x = –5

  4. 10.5 The Quadratic Formula Use the quadratic formula to solve: –15x2 – 21x – 6 = 0 a = –15, b = –21, c = –6 √(-21)2 + (-4)(-15)(-6) –(–21)  21  9 2(–15) –30 √(441) + (-360) 21  12 30 = = -30 -30 –30 √81 21  -2 x = -1 x = 5 –30

  5. –b b2 – 4ac x = 2a 10.5 The Quadratic Formula A rocket is launched from atop a 45-foot cliff with an initial velocity of 89 feet per second. The height of the rocket ‘t’ seconds after launch is given by the following equation: h = –16t2 + 89t + 45. Use the quadratic formula to find out how long after the rocket is launched, it will hit the ground. Round your answer to the nearest tenth of a second. (6.02899) 6.0 seconds a = b = c = -16 89 45

  6. –bb2 + (– 4ac) x = 2a x = –4  64 2 10.5 The Quadratic Formula Use the quadratic formula to solve: x2 + 4x – 12 = 0 a = 1, b = 4, c = –12 x = –(4) √(4)2+(–4)(1)(–12) 2(1) –4 – 8 –4 + 8 = = 2 2 4 –12 = = 2 2 –4 ± 8 x = 2 x = 2 x = –6

  7. –bb2 + (– 4ac) –(–9)  (–9)2 + (– 4)(2)(5) x = x = 2a 2(2) x = 9  41 4 10.5 The Quadratic Formula Use the quadratic formula to solve: 2x2 – 9x + 5 = 0 a = 2, b = –9, c = 5 9 – 6.4 9 + 6.4 = = 4 4 2.6 15.4 = = 4 4 9 ± 6.4 x = x = 3.85 4 x = 0.65

  8. –b b2 – 4ac x = 2a 10.5 The Quadratic Formula A rocket is launched from atop a 42-foot cliff with an initial velocity of 122 feet per second. The height of the rocket ‘t’ seconds after launch is given by the following equation: h = –16t2 + 122t + 42. Use the quadratic formula to find out how long after the rocket is launched, it will hit the ground. Round your answer to the nearest tenth of a second. (7.95498) 8.0 seconds a = b = c = -16 122 42

  9. 10.5 The Quadratic Formula Discriminant For ax2 + bx + c = 0, where a 0: b2 + (–4ac) Discriminant value (Dv): If the Dv is less than 0 – Quad Eq has NO solution If the Dv IS zero – the Quad Eq has only ONE solution If the Dv is greater than 0, the Quad Eq has TWO solutions

  10. 10.5 The Quadratic Formula b2 + (–4ac) x2 + 5x – 8 = 0 a = 1, b = 5, c = –8 Use the discriminant to find the # of real solutions: discriminant: 52 + (–4)(1)(–8) = 57 two solutions a = 4, b = –4, c = 1 4x2 – 4x + 1 = 0 one solution discriminant: (–4)2 + (–4)(4)(1) = 0 a = 3, b = 4, c = –12 3x2 + 4x – 12 = 0 two solutions discriminant: 42 + (–4)(3)(–12) = 160

  11. –bb2 + (– 4ac) x = 2a x = 3  121 2 10.5 The Quadratic Formula Use the quadratic formula to solve: x2– 3x – 28 = 0 a = 1, b = –3, c = –28 x = –(–3) √(–3)2+(–4)(1)(–28) 2(1) 3 – 11 3 + 11 = = 2 2 14 –8 = = 2 2 3 ± 11 x = 2 x = 7 x = –4

  12. –bb2 + (– 4ac) –2  (2)2 + (– 4)(3)(–4) x = x = 2a 2(3) x = –2  52 6 10.5 The Quadratic Formula Use the quadratic formula to solve: 3x2 + 2x – 4 = 0 a = 3, b = 2, c = –4 –2 – 7.2 –2 + 7.2 = = 6 6 5.2 –9.2 = = –2 ± 7.2 6 6 x = 6 x = 0.87 x =–1.53

  13. –b b2 – 4ac x = 2a 10.5 The Quadratic Formula A rocket is launched from atop a 30-foot cliff with an initial velocity of 80 feet per second. The height of the rocket ‘t’ seconds after launch is given by the following equation: h = –16t2 + 80t + 30. Use the quadratic formula to find out how long after the rocket is launched, it will hit the ground. Round your answer to the nearest tenth of a second. (5.34044) 5.4 seconds a = b = c = -16 80 30

  14. –bb2 + (– 4ac) x = 2a x = 6   8 2 10.5 The Quadratic Formula Use the quadratic formula to solve: x2– 6x + 7 = 0 a = 1, b = –6, c = 7 x = –(–6) √(–6)2+(–4)(1)(7) 2(1) 6 – 2.83 6 + 2.83 = = 2 2 8.83 3.17 = = 2 2 6 ± 2.83 x = 2 x = 4.42 x = 1.59

  15. –bb2 + (– 4ac) x = 2a x = 15  81 18 10.5 The Quadratic Formula Use the quadratic formula to solve: 9x2– 15x + 4 = 0 a = 9, b = –15, c = 4 x = –(–15) √(–15)2+(–4)(9)(4) 2(9) 15 – 9 15 + 9 = = 18 18 24 6 = = 18 18 15 ± 9 x = 18 x = 1.33 x = 0.33

  16. –b b2 – 4ac x = 2a 10.5 The Quadratic Formula A rocket is launched from atop a 102-foot cliff with an initial velocity of 95 feet per second. The height of the rocket ‘t’ seconds after launch is given by the following equation: h = –16t2 + 95t + 102. Use the quadratic formula to find out how long after the rocket is launched, it will hit the ground. Round your answer to the nearest tenth of a second. (6.86598) 6.9 seconds a = b = c = -16 95 102

  17. 10.5 The Quadratic Formula a = 3, b = –2, c = 1 3x2 – 2x + 1 = 0 Use the discriminant to find the # of real solutions: No solution discriminant: (–2)2 + (–4)(3)(1) = –8 a = –4, b = 6, c = –2 –4x2 + 6x – 2 = 0 two solutions discriminant: 62 + (–4)(–4)(–2) = 4 x2– 2x + 3 = 0 a = 1, b = –2, c = 3 No solution discriminant: (–2)2 + (–4)(1)(3) = –8

  18. –bb2 + (– 4ac) x = 2a x = 10   8 2 10.5 The Quadratic Formula Use the quadratic formula to solve: x2– 10x + 23 = 0 a = 1, b = –10, c = 23 x = –(–10) √(–10)2+(–4)(1)(23) 2(1) 10 – 2.83 10 + 2.83 = = 2 2 12.83 7.17 = = 2 2 10 ± 2.83 x = 2 x = 6.42 x = 3.59

  19. –bb2 + (– 4ac) –(4)  (4)2 + (–4)(3)(–15) x = x = 2a 2(3) x = –4  196 6 10.5 The Quadratic Formula Use the quadratic formula to solve: 3x2 + 4x – 15 = 0 a = 3, b = 4, c = –15 –4 – 14 –4 + 14 = = 6 6 –18 10 = = 6 6 –4 ± 14 x = x =–3 x = 1.67 6

  20. –bb2 + (– 4ac) x = 2a x = 4  12 2 10.5 The Quadratic Formula Use the quadratic formula to solve: x2– 4x + 1 = 0 a = 1, b = –4, c = 1 x = –(–4) √(–4)2+(–4)(1)(1) 2(1) 4 – 3.46 4 + 3.46 = = 2 2 7.46 0.54 = = 2 2 4 ± 3.46 x = 2 x = 3.73 x = 0.27

  21. –(–14)  (–14)2 + (– 4)(2)(20) x = 2(2) x = 14  36 4 10.5 The Quadratic Formula Use the quadratic formula to solve: 2x2 – 14x + 20 = 0 a = 2, b = –14, c = 20 14 – 6 14 + 6 = = 4 4 8 20 = = 4 4 14 ± 6 x = x = 5 x = 2 4

  22. –(37)  (37)2 + (– 4)(14)(-42) x = 2(14) x = -37  3721 28 10.5 The Quadratic Formula Use the quadratic formula to solve: 14x2 + 37x – 42 = 0 a = 14, b = 37, c = –42 -37 – 61 -37 + 61 = = 28 28 24 -98 = = -37 ± 61 28 28 x = 28 6 -7 x = x = 7 2

  23. –(–14)  (–14)2 + (– 4)(2)(20) x = 2(2) x = 14  36 4 10.5 The Quadratic Formula Factor each expression: 2x2 – 14x + 20 = 0 a = 2, b = –14, c = 20 14 – 6 14 + 6 = = 4 4 8 20 = = 4 4 14 ± 6 x = x = 5 x = 2 4

  24. –(–8)  (–8)2 + (– 4)(6)(-14) x = 2(6) x = 8  400 12 10.5 The Quadratic Formula Use the quadratic formula to find the factors: 6x2 – 8x – 14 = a = 6, b = –8, c = –14 8 – 20 8 + 20 = = 12 12 28 -12 = = 12 12 8 ± 20 7 x = x = 12 x = -1 3

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