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Maths Gcse Aqa Higher

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  1. Scatter-Graphs-And-Correlation Aqa Higher
    2 主题
  2. Cumulative-Frequency-And-Box-Plots Aqa Higher
    4 主题
  3. Histograms Aqa Higher
    3 主题
  4. Statistical-Diagrams Aqa Higher
    5 主题
  5. Averages-Ranges-And-Data Aqa Higher
    7 主题
  6. Combined-And-Conditional-Probability Aqa Higher
    3 主题
  7. Tree-Diagrams Aqa Higher
    1 主题
  8. Simple-Probability-Diagrams Aqa Higher
    3 主题
  9. Transformations Aqa Higher
    5 主题
  10. Vectors Aqa Higher
    6 主题
  11. 3D-Pythagoras-And-Trigonometry Aqa Higher
    1 主题
  12. Sine-Cosine-Rule-And-Area-Of-Triangles Aqa Higher
    4 主题
  13. Pythagoras-And-Trigonometry Aqa Higher
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  14. Area-And-Volume-Of-Similar-Shapes Aqa Higher
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  15. Congruence-Similarity-And-Geometrical-Proof Aqa Higher
    5 主题
  16. Volume-And-Surface-Area Aqa Higher
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  17. Circles-Arcs-And-Sectors Aqa Higher
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  18. Area-And-Perimeter Aqa Higher
    4 主题
  19. Circle-Theorems Aqa Higher
    7 主题
  20. Bearings-Scale-Drawing-Constructions-And-Loci Aqa Higher
    5 主题
  21. Angles-In-Polygons-And-Parallel-Lines Aqa Higher
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  22. Symmetry-And-Shapes Aqa Higher
    6 主题
  23. Exchange-Rates-And-Best-Buys Aqa Higher
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  24. Standard-And-Compound-Units Aqa Higher
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  25. Direct-And-Inverse-Proportion Aqa Higher
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  26. Problem-Solving-With-Ratios Aqa Higher
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  27. Ratios Aqa Higher
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  28. Sequences Aqa Higher
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  29. Transformations-Of-Graphs Aqa Higher
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  30. Graphing-Inequalities Aqa Higher
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  31. Solving-Inequalities Aqa Higher
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  32. Real-Life-Graphs Aqa Higher
    4 主题
  33. Estimating-Gradients-And-Areas-Under-Graphs Aqa Higher
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  34. Equation-Of-A-Circle Aqa Higher
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  35. Functions Aqa Higher
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  36. Forming-And-Solving-Equations Aqa Higher
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  37. Graphs-Of-Functions Aqa Higher
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  38. Linear-Graphs Aqa Higher
    4 主题
  39. Coordinate-Geometry Aqa Higher
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  40. Iteration Aqa Higher
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  41. Simultaneous-Equations Aqa Higher
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  42. Quadratic-Equations Aqa Higher
    4 主题
  43. Linear-Equations Aqa Higher
    1 主题
  44. Algebraic-Proof Aqa Higher
    1 主题
  45. Rearranging-Formulas Aqa Higher
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  46. Algebraic-Fractions Aqa Higher
    4 主题
  47. Completing-The-Square Aqa Higher
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  48. Factorising Aqa Higher
    6 主题
  49. Expanding-Brackets Aqa Higher
    3 主题
  50. Algebraic-Roots-And-Indices Aqa Higher
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  51. Using-A-Calculator Aqa Higher
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  52. Surds Aqa Higher
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  53. Rounding-Estimation-And-Bounds Aqa Higher
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  54. Fractions-Decimals-And-Percentages Aqa Higher
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  55. Introduction Aqa Higher
    7 主题
  56. Simple-And-Compound-Interest-Growth-And-Decay Aqa Higher
    4 主题
  57. Percentages Aqa Higher
    3 主题
  58. Fractions Aqa Higher
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  59. Powers-Roots-And-Standard-Form Aqa Higher
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  60. Prime-Factors-Hcf-And-Lcm Aqa Higher
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  61. Number-Operations Aqa Higher
    10 主题
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Exam code:8300

Solving by completing the square

How do I solve a quadratic equation by completing the square?

  • To solve x2 + bx + c = 0 

    • replace the first two terms, x2 + bx, with (x + p)2 – p2 where p is half of b

    • This is completing the square

      • x2 + bx + c = 0 becomes (x + p)2 – p2 + c = 0

      • (where p is half of b)

    • rearrange this equation to make x the subject (using ±√)

  • For example, solve x2 + 10x + 9 = 0 by completing the square

    • x2 + 10x becomes (x + 5)2 – 52

    • so x2 + 10x + 9 = 0 becomes (x + 5)2 – 52 + 9 = 0

    • make x the subject (using ±√)

      • (x + 5)2 – 25 + 9 = 0

      • (x + 5)2 = 16

      • x + 5 = ±√16

      • x + 5 = ±4

      • x = -5 ±4

      • x = -1 or x = -9

  • It also works with numbers that lead to surds

    • The answers found will be in exact (surd) form

Examiner Tips and Tricks

  • When making x the subject to find the solutions, don’t expand the squared bracket back out again!

    •  Remember to use ±√ to get two solutions

How do I solve by completing the square when there is a coefficient in front of the x2 term?

  • If the equation is ax2 + bx + c = 0 with a number (other than 1) in front of x2

    • you can divide both sides by a first (before completing the square)

      • For example 3x2 + 12x + 9 = 0

      • Divide both sides by 3

        • x2 + 4x + 3 = 0

      • Complete the square on this easier equation

  • This trick only works when completing the square to solve a quadratic equation

    • i.e. it has an “=0” on the right-hand side

  • Don’t do this when using completing the square to rewrite a quadratic expression in a new form

    • i.e. when there is no “=0”

    • For that, you must factorise out the a (but not divide by it)

      • a x squared plus b x plus c equals a open square brackets x squared plus b over a x close square brackets plus c and so on

  • The quadratic formula actually comes from completing the square to solve ax2 + bx + c = 0

    • a, b and c are left as letters when completing the square

      • This makes it as general as possible

  • You can see hints of this when you solve quadratics 

    • For example, solving x2 + 10x + 9 = 0 

      • by completing the square, (x + 5)2 = 16 so x = -5 ± 4 (as above) 

      • by the quadratic formula, x equals fraction numerator negative 10 plus-or-minus square root of 64 over denominator 2 end fraction equals negative 5 plus-or-minus 8 over 2 = -5 ± 4 (the same structure)

Worked Example

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