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Objective: The learner will..,

10.4 Solving Quadratic Equations . Objective: The learner will..,. NCSCOS. Solve quadratic equations by factoring, completing the square finding the vertex, AOS, and zeros. Find the two ordered pairs for the intersection of the parabola and the linear equation. 1.01, 4.02.

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Objective: The learner will..,

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  1. 10.4 Solving Quadratic Equations Objective: The learner will.., NCSCOS • Solve quadratic equations by factoring, completing the square finding the vertex, AOS, and zeros. • Find the two ordered pairs for the intersection of the parabola and the linear equation. • 1.01, 4.02

  2. 10.1 Graphing Parabolas Factor, find the AOS, and the Vertex: y = x2– 6x + 8 x2– 6x + 8 = 0 x2– 6x + 8 = 0 (x– 2)(x– 4) = 0 (3)2– 6(3) + 8 = 0 x = 2 x = 4 (x– 2) = 0 (x– 4) = 0 -1 9 – 18 + 8 = Find the AOS: vertex: (3, ) -1 2 + 4 Find the vertex form: =3 x = 2 y = (x – 3)2–1 AOS:3

  3. 10.1 Graphing Parabolas Factor, find the AOS, and the Vertex: y = x2 + 2x – 24 x2 + 2x –24 = 0 x2 + 2x – 24 = 0 (x + 6)(x– 4) = 0 (–1)2 + 2(–1) – 24 = 0 x = 4 x = –6 (x– 4) = 0 (x + 6) = 0 –25 1 – 2 – 24 = Find the AOS: –25 vertex: (-1, ) 4 – 6 Find the vertex form: =–1 x = 2 y = (x+1)2–25 AOS:–1

  4. 10.1 Graphing Parabolas Factor, find the AOS, and the Vertex: y = x2– 6x + 5 x2– 6x + 5 = 0 x2– 6x + 5 = 0 (3)2– 6(3) + 5 = 0 (x– 1)(x– 5) = 0 -4 9 – 18 + 5 = x = 1 x = 5 (x– 1) = 0 (x– 5) = 0 vertex: (3, ) -4 Find the AOS: Find the vertex form: 1 + 5 =3 x = 2 y = (x – 3)2–4 AOS:3

  5. 10.4 Solving Quadratic Equations Complete the Square to find the vertex, AOS, and zeros: y = x2 – 4x + 1 y = x2 – 6x + 7 (x2 – 4x+ 4) + 1 – 4 = 0 (x2 – 6x+ 9) + 7 – 9 = 0 (x – 2)2 – 3 = 0 (x – 3)2 – 2 = 0 Vrtx: (2, –3) AOS: Vrtx: 2 (3, –2) 3 AOS: (x – 2)2 – 3 = 0 (x – 3)2 – 2 = 0 (x – 2)2 = 3 (x – 3)2 = 2 x – 2 = ± 1.73 x – 3 = ± 1.41 x = 3.7 & .27 x = 4.41 & 1.59

  6. 10.4 Solving Quadratic Equations Complete the Square to find the vertex, AOS, and zeros: y = x2 – 8x + 3 y = x2 – 10x + 23 (x2 – 8x + 16) + 3 – 16 = 0 (x2 – 10x + 25) + 23 – 25 = 0 (x – 4)2 – 13 = 0 (x – 5)2 – 2 = 0 Vrtx: (4, -13) AOS: 4 (5, -2) Vrtx: 5 AOS: (x – 4)2 – 13 = 0 (x – 5)2 – 2 = 0 (x – 4)2 = 13 (x – 5)2 = 2 x – 4 = ± 3.61 (x – 5) = ±1.41 x = 7.6 & .39 x = 6.4 & 3.6

  7. 10.4 Solving Quadratic Equations Complete the Square to find the vertex, AOS, and zeros: x2 – 8x + 3 = 0 x2 + 6x + 4 = 0 (x2 – 8x + 16) + 3 – 16 = 0 (x2 + 6x + 9) + 4 – 9 = 0 (x – 4)2 – 13 = 0 (x + 3)2 – 5 = 0 4 (4, –13) AOS: Vrtx: -3 (-3, -5) Vrtx: AOS: (x – 4)2 – 13 = 0 (x + 3)2 – 5 = 0 (x – 4)2 = 13 (x + 3)2 = 5 x – 4 =  3.61 x + 3 =  2.24 x = .39 & 7.6 x = -0.76 & -5.24

  8. 10.4 Solving Quadratic Equations Complete the Square to find the vertex, AOS, and zeros: x2 + 2x – 7 = 0 x2 – 4x – 1 = 0 (x2 + 2x + 1) – 7 – 1 = 0 (x2 – 4x + 4) – 1 – 4 = 0 (x + 1)2 – 8 = 0 (x – 2)2 – 5 = 0 -1 Vrtx: (2, -5) (-1, -8) 2 Vrtx: AOS = AOS: (x – 2)2 – 5 = 0 (x + 1)2 – 8 = 0 (x – 2)2 = 5 (x + 1)2 = 8 x – 2 =  2.24 x + 1 =  2.83 x = -0.24 & 4.24 x = -3.83 & 1.83

  9. 10.4 Solving Quadratic Equations Find the two ordered pairs for the intersection of the parabola and the linear equation. y = x2 + 3x – 15 and y = -5 x2 + 3x – 15 = -5 x2 + 3x – 10 = 0 y = -5 -5 (-5, ) (x + 5)(x – 2) = 0 (x + 5) = 0 x = -5 y = -5 (x – 2) = 0 x = 2 (2, ) -5

  10. 10.4 Solving Quadratic Equations Find the vertex, AOS, and x-intercepts: y = x2 + 3x – 15 and y = -5 (x2 + 3x + 2.25) – 15 – 2.25 (x + 1.5)2 – 17.25 Vrtx: (–1.5, –17.25) AOS: –1.5 x2 + 3x – 15 = 0 (x + 5)(x – 3) = 0 (2,-5) (-5,-5) x = –5 & 3 (2, -5) (-5, -5)

  11. 10.4 Solving Quadratic Equations Find the two ordered pairs for the intersection of the parabola and the linear equation. y = x2 – 10x + 21 and y = x – 3 y = x – 3 x2 – 10x + 21 = x – 3 y = 8 – 3 x2 – 11x + 24 = 0 y = 5 (x – 8)(x – 3) = 0 5 (8, ) (x – 8) = 0 x = 8 y = x – 3 x = 3 (x – 3) = 0 y = 3 – 3 y = 0 0 (3, )

  12. 10.4 Solving Quadratic Equations Find vertex, AOS, and x-intercepts then graph the parabola and linear equation: y = x2 – 10x + 21 and y = x – 3 (x2 – 10x + 25) + 21 – 25 (x – 5)2 + 21 – 25 (x – 5)2 – 4 = 0 Vrtx: (5, –4) (8,5) AOS: 5 (x – 5)2 = 4 (3,0) x – 5 =  2 x = 7 & 3 (3, 0) (8, 5)

  13. 10.4 Solving Quadratic Equations Find the two ordered pairs for the intersection of the parabola and the linear equation. y = x2 – 4x + 1 and y = -2x + 4 y = -2x + 4 x2 – 4x + 1 = -2x + 4 y = -2(-1) + 4 x2 – 4x + 2x + 1 – 4 = 0 y = 6 x2 – 2x – 3 = 0 (-1, ) 6 (x + 1)(x – 3) = 0 y = -2x + 4 (x + 1) = 0 x = -1 y = -2(3) + 4 x = 3 (x – 3) = 0 y = -2 -2 (3, )

  14. 10.4 Solving Quadratic Equations Find vertex, AOS, and x-intercepts then graph the parabola and linear equation: y = x2 – 4x + 1 y = -2x + 4 (x2 – 4x + 4) + 1 – 4 (x – 2)2 + 1 – 4 (-1,6) (x – 2)2 – 3 = 0 V (2, –3) (x – 2)2 = 3 (3,-2) x – 2 =  1.73 x = 3.73 & .27 (-1, 6) (3, -2)

  15. 10.4 Solving Quadratic Equations Find the two ordered pairs for the intersection of the parabola and the linear equation. y = x2 – 4x + 11 and y = x + 5 y = x + 5 x2 – 4x + 11 = x + 5 y = (2) + 5 x2 – 5x + 6 = 0 y = 7 (2, ) 7 (x – 2)(x – 3) = 0 y = x + 5 (x – 2) = 0 x = 2 y = (3) + 5 (x – 3) = 0 x = 3 y = 8 (3, ) 8

  16. 10.4 Solving Quadratic Equations Find vertex, AOS, and x-intercepts then graph the parabola and linear equation: y = x2 – 4x + 11 and y = x + 5 (x2 – 4x + 4) + 11 – 4 (x – 2)2 + 11 – 4 (x – 2)2 + 7 = 0 Vrtx: (2, 7) (3,8) AOS: 2 (2,7) (x – 2)2 = –7 Can’t take the sq root; no x-intercepts (2, 7) (3, 8)

  17. 10.4 Solving Quadratic Equations Find the two ordered pairs for the intersection of the parabola and the linear equation. y = x2 – 7x + 10 and y = x – 2 y = x – 2 x2 – 7x + 10 = x – 2 y = 6 – 2 x2 – 8x + 12 = 0 y = 4 (x – 6)(x – 2) = 0 4 (6, ) (x – 6) = 0 x = 6 y = x – 2 x = 2 (x – 2) = 0 y = 2 – 2 y = 0 0 (2, )

  18. 10.4 Solving Quadratic Equations Find vertex, AOS, and x-intercepts then graph the parabola and linear equation: y = x2 – 7x + 10 and y = x – 2 (x2 – 7x + 12.25) + 10 – 12.25 (x – 3.5)2 + 10 – 12.25 (x – 3.5)2 – 2.25 = 0 Vrtx: (3.5, –2.25) AOS: 3.5 (x – 3.5)2 = 2.25 x – 3.5 =  1.5 x = 5 & 2 (2, 0) (6, 4) (2,0) (6,4)

  19. 10.4 Solving Quadratic Equations Find the two ordered pairs for the intersection of the parabola and the linear equation. y = x2 – x – 9 and y = –2x – 3 (–3, 3) and (2, –7) y = x2 – 4x + 3 and y = x + 3 (0, 3) and (5, 8) y = x2 – 8x + 5 and y = 2x – 4 (1, –2) and (9, 14)

  20. 10.4 Solving Quadratic Equations Find the two ordered pairs for the intersection of the parabola and the linear equation. y = x2 + 6x + 8 and y = 2x + 8 (–4, 0) and (0, 8) y = x2 + 4x – 5 and y = −2x – 5 (−6, 7) and (0, −5) y = x2 – 6x + 9 and y = x – 3 (3, 0) and (4, 1)

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