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Form the quadratic equations with the following roots:

1. Form the quadratic equations with the following roots:. 2 and 4. ( i ). Roots x = 2 and x = 4. x – 2 = 0 and x – 4 = 0. ( x – 2)( x – 4) = 0. x ( x – 4) – 2( x – 4) = 0. x 2 – 4 x – 2 x + 8 = 0. x 2 – 6 x + 8 = 0. 1.

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Form the quadratic equations with the following roots:

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  1. 1. Form the quadratic equations with the following roots: 2 and 4 (i) Roots x = 2 and x = 4 x – 2 = 0 and x – 4 = 0 (x – 2)(x – 4) = 0 x(x – 4) – 2(x – 4) = 0 x2 – 4x – 2x + 8 = 0 x2 – 6x + 8 = 0

  2. 1. Form the quadratic equations with the following roots: 7 and 3 (ii) Roots x = 7 and x = 3 x – 7 = 0 and x – 3 = 0 (x – 7)(x – 3) = 0 x(x – 3) – 7(x – 3) = 0 x2 – 3x – 7x + 21 = 0 x2 – 10x + 21 = 0

  3. 1. Form the quadratic equations with the following roots: 4 and 6 (iii) Roots x = 4 and x = 6 x – 4 = 0 and x – 6 = 0 (x – 4)(x – 6) = 0 x(x – 6) – 4(x – 6) = 0 x2 – 6x – 4x + 24 = 0 x2 – 10x + 24 = 0

  4. 1. Form the quadratic equations with the following roots: − 1 and 3 (iv) Roots x = – 1 and x = 3 x + 1 = 0 and x – 3 = 0 (x + 1)(x – 3) = 0 x(x – 3) + 1(x – 3) = 0 x2 – 3x + x – 3 = 0 x2 – 2x – 3 = 0

  5. 1. Form the quadratic equations with the following roots: – 2 and 9 (v) Roots x = – 2 and x = 9 x + 2 = 0 and x – 9 = 0 (x + 2)(x – 9) = 0 x(x – 9) + 2(x – 9) = 0 x2 – 9x + 2x – 18 = 0 x2 – 7x – 18 = 0

  6. 1. Form the quadratic equations with the following roots: 8 and − 2 (vi) Roots x = 8 and x = – 2 x – 8 = 0 and x + 2 = 0 (x – 8)(x + 2) = 0 x(x + 2) – 8(x + 2) = 0 x2 + 2x – 8x – 16 = 0 x2 – 6x – 16 = 0

  7. 1. Form the quadratic equations with the following roots: 12 and − 3 (vii) Roots x = 12 and x = – 3 x – 12 = 0 and x + 3 = 0 (x – 12)(x + 3) = 0 x(x + 3) – 12(x + 3) = 0 x2 + 3x – 12x – 36 = 0 x2 – 9x – 36 = 0

  8. 1. Form the quadratic equations with the following roots: − 2 and − 6 (viii) Roots x = – 2 and x = – 6 x + 2 = 0 and x + 6 = 0 (x + 2)(x + 6) = 0 x(x + 6) + 2(x + 6) = 0 x2 + 6x + 2x + 12 = 0 x2 + 8x + 12 = 0

  9. 1. Form the quadratic equations with the following roots: − 7 and − 4 (ix) Roots x = – 7 and x = – 4 x + 7 = 0 and x + 4 = 0 (x + 7)(x + 4) = 0 x(x + 4) + 7(x + 4) = 0 x2 + 4x + 7x + 28 = 0 x2 + 11x + 28 = 0

  10. 2. The quadratic equation x2 + px+ q = 0 has roots 7 and 2. Find the values of pand q. x2 + px + q = 0 Roots x = 7 and x = 2 x – 7 = 0 and x – 2 = 0 (x – 7)(x – 2) = 0 x(x – 2) – 7(x – 2) = 0 x2 – 2x – 7x + 14 = 0 x2 – 9x + 14 = 0 p = – 9, q = 14

  11. 3. The quadratic equation x2 + cx + d = 0 has roots − 6 and − 4. Find the values of candd. x2 + cx + d = 0 Roots x = – 6 and x = – 4 x + 6 = 0 and x + 4 = 0 (x + 6)(x + 4) = 0 x(x + 4) + 6(x + 4) = 0 x2 + 4x + 6x + 24 = 0 x2 + 10x + 24 = 0 c = 10, d = 24

  12. 4. (i) Form the quadratic equation with roots − 3 and 8. Roots x = − 3 and x = 8 x + 3 = 0 and x – 8 = 0 (x + 3)(x – 8) = 0 x(x – 8) + 3(x – 8) = 0 x2 – 8x + 3x – 24 = 0 x2 – 5x – 24 = 0

  13. 4. (ii) Verify your result by factorising and solving this quadratic equation. x2 – 5x – 24 = 0 (x – 8)(x + 3) = 0 x – 8 = 0 and x + 3 = 0 x = 8 and x = –3

  14. 5. For each of the following graphs: identify the roots of the function (a) (i) Roots x = – 3 and x = 1

  15. 5. For each of the following graphs: form the equation (a) (ii) x = – 3 and x = 1 x + 3 = 0 and x – 1 = 0 (x + 3)(x – 1) = 0 x(x – 1) + 3(x – 1) = 0 x2 – x + 3x – 3 = 0 x2 + 2x – 3 = 0

  16. 5. For each of the following graphs: verify the roots of the equation, algebraically (a) (iii) x = – 3 x2 + 2x – 3 = 0 (– 3)2 + 2(– 3) – 3 = 0 9 – 6 – 3 = 0 0 = 0 x = 1 12 + 2(1) – 3 = 0 1 + 2 – 3 = 0 0 = 0

  17. 5. For each of the following graphs: given each graph is illustrating a function f (x), write the function in each case. (a) (iv) U-shape graph, so f (x) = x2 + 2x – 3

  18. 5. For each of the following graphs: identify the roots of the function (b) (i) Roots x = – 4 and x = 2

  19. 5. For each of the following graphs: form the equation (b) (ii) x = – 4 and x = 2 x + 4 = 0 and x – 2 = 0 (x + 4)(x – 2) = 0 x(x – 2) + 4(x – 2) = 0 x2 – 2x + 4x – 8 = 0 x2 + 2x – 8 = 0

  20. 5. For each of the following graphs: verify the roots of the equation, algebraically (b) (iii) x = – 4 x2 + 2x – 8 = 0 (– 4)2 + 2(– 4) – 8 = 0 16 – 8 – 8 = 0 0 = 0 x = 2 (2)2 + 2(2) – 8 = 0 4 + 4 – 8 = 0 0 = 0

  21. 5. For each of the following graphs: given each graph is illustrating a function f (x), write the function in each case. (b) (iv) The graph is  shape therefore x2 must be negative, so the equation is multiplied by – 1 x2 + 2x – 8 = 0 (multiply all parts by – 1) – x2 – 2x + 8 = 0 f(x) = – x2 – 2x + 8

  22. 5. For each of the following graphs: identify the roots of the function (c) (i) Roots x = 5 and x = 5

  23. 5. For each of the following graphs: form the equation (c) (ii) Roots x = 5 and x = 5 x – 5 = 0 and x – 5 = 0 (x – 5)(x – 5) = 0 x(x – 5) – 5(x – 5) = 0 x2 – 5x – 5x + 25 = 0 x2 – 10x + 25 = 0

  24. 5. For each of the following graphs: verify the roots of the equation, algebraically (c) (iii) x = 5 x2 – 10x + 25 = 0 (5)2 – 10(5) + 25 = 0 25 – 50 + 25 = 0 0 = 0

  25. 5. For each of the following graphs: given each graph is illustrating a function f (x), write the function in each case. (c) (iv) The graph is  shape, therefore x2 must be negative, so the equation is multiplied by – 1 x2 – 10x + 25 = 0 (multiply all parts by – 1) – x2 + 10x – 25 = 0 f(x) = –x2 + 10x – 25

  26. 5. For each of the following graphs: identify the roots of the function (d) (i) Roots x = 1·5 and x = 6

  27. 5. For each of the following graphs: form the equation (ii) (d) x = 1·5 and x = 6 2x = 3 x – 6 = 0 2x – 3 = 0 (2x – 3)(x – 6) = 0 2x(x – 6) – 3(x – 6) = 0 2x2 – 12x – 3x + 18 = 0 2x2 – 15x + 18 = 0

  28. 5. For each of the following graphs: verify the roots of the equation, algebraically (iii) (d) x = 1·5 2x2 – 15x + 18 = 0 2(1·5)2 – 15(1·5) + 18 = 0 2(2·25) – 22·5 + 18 = 0 4·5 – 22·5 + 18 = 0 0 = 0

  29. 5. For each of the following graphs: verify the roots of the equation, algebraically (iii) (d) x = 6 2x2 – 15x + 18 = 0 2(6)2 – 15(6) + 18 = 0 2(36) – 90 + 18 = 0 72 – 90 + 18 = 0 0 = 0

  30. 5. For each of the following graphs: given each graph is illustrating a function f (x), write the function in each case. (iv) (d) U-shape graph, so f(x) = 2x2 – 15x + 18

  31. 6. (i) Form the quadratic equation with roots 5 and − 9. Roots x = 5 and x = – 9 x – 5 = 0 and x + 9 = 0 (x – 5)(x + 9) = 0 x(x + 9) – 5(x + 9) = 0 x2 + 9x – 5x – 45 = 0 x2 + 4x – 45 = 0

  32. 6. (ii) Verify your result by factorising and solving this quadratic equation. x2 + 4x – 45 = 0 (x + 9)(x – 5) = 0 x + 9 = 0 and x – 5 = 0 x = – 9 x = 5

  33. 6. (iii) The graph of a function, k (x), has roots 5 and − 9 and crosses the y-axis at the point (0, − 45). Draw a sketch of the graph of k(x).

  34. 6. (iv) Write down the function, k (x). k(x) = x2 + 4x – 45

  35. 7. (i) Form the quadratic equation with roots − 4 and − 11. Roots: x = – 4 and x = – 11 x + 4 = 0 and x + 11 = 0 (x + 4)(x + 11) = 0 x(x + 11) + 4(x + 11) = 0 x2 + 11x + 4x + 44 = 0 x2 + 15x + 44 = 0

  36. 7. (ii) Verify your result by factorising and solving this quadratic equation. x2 + 15x + 44 = 0 (x + 11)(x + 4) = 0 x + 11 = 0 and x + 4 = 0 x = –11 and x = – 4

  37. 7. The graph of a function, g (x), has roots − 4 and − 11 and crosses the y-axis at the point (0, − 44). Draw a sketch of the graph of g (x).

  38. 7. (iv) Write down the function, g(x). To cross y-axis at y = − 44, graph is  shaped, so must have a negative x2. Multiply the equation by – 1 to give: g(x) = – x2 – 15x – 44.

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