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GCSEMathsAQAHigherFlashcards2. AlgebraCompleting the Square

Completing the Square (AQA GCSE Maths)

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  • FrontCompleting the Square

    When completing the square for the expression x squared plus b x plus c, explain how to find the value of p in the expression open parentheses x plus p close parentheses squared plus q.

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  • When completing the square for the expression x squared plus b x plus c, explain how to find the value of p in the expression open parentheses x plus p close parentheses squared plus q.

    Completing the square for x squared plus b x plus c gives the form open parentheses x plus p close parentheses squared plus q.

    The value of p is half of the value of b.

  • True or False?

    The coordinates of the turning point (vertex) of the quadratic curve y equals open parentheses x plus 3 close parentheses squared plus 5 are open parentheses 3 comma space 5 close parentheses.

    False.

    The coordinates of the turning point (vertex) of the quadratic curve y equals open parentheses x plus 3 close parentheses squared plus 5 are not open parentheses 3 comma space 5 close parentheses.

    The correct coordinates are open parentheses negative 3 comma space 5 close parentheses.

    This is a common mistake in the exam! If y equals open parentheses x plus p close parentheses squared plus q is the curve, then open parentheses negative p comma space q close parentheses are the coordinates of the turning point.

  • What is the first step to completing the square of the quadratic expression a x squared plus b x plus c where a is not equal to 1, e.g. 3 x squared plus 12 x minus 8?

    The first step to completing the square of the quadratic expression a x squared plus b x plus c where a is not equal to 1 is to factorise out a.

    This gives a open square brackets x squared plus b over a x plus c over a close square brackets.

    You can then complete the square inside the big brackets.

    E.g. to complete the square of the expression 3 x squared plus 12 x minus 8, the first step would be to factorise out 3, giving 3 open square brackets x squared plus 4 x minus 8 over 3 close square brackets.

  • True or False?

    The coordinates of the turning points on the curves y equals open parentheses x plus 2 close parentheses squared plus 4 and y equals 3 open parentheses x plus 2 close parentheses squared plus 4 are the same.

    True.

    The coordinates of the turning points on the curves y equals open parentheses x plus 2 close parentheses squared plus 4 and y equals 3 open parentheses x plus 2 close parentheses squared plus 4 are the same.

    The coordinates of the turning point on y equals a open parentheses x plus p close parentheses squared plus q are always open parentheses negative p comma space q close parentheses, regardless of the value of a (even if a less than 0).

  • Explain how writing x squared minus 4 x plus 9 in the form open parentheses x minus 2 close parentheses squared plus 5 shows that any output of the function straight f open parentheses x close parentheses equals x squared minus 4 x plus 9 is always greater than or equal to 5.

    If x squared minus 4 x plus 9 can be written as open parentheses x minus 2 close parentheses squared plus 5 by completing the square, then straight f open parentheses x close parentheses equals x squared minus 4 x plus 9 can be written as straight f open parentheses x close parentheses equals open parentheses x minus 2 close parentheses squared plus 5.

    The coordinates of the turning point will be open parentheses 2 comma space 5 close parentheses.

    As this is a positive quadratic, the turning point will be a minimum and the therefore the value of straight f open parentheses x close parentheses will always be greater than or equal to 5.

Previous:FactorisingNext:Algebraic Fractions

    1. Topic 1

      Number Operations

    2. Topic 2

      Prime Factors, HCF & LCM

    3. Topic 3

      Powers, Roots & Standard Form

    4. Topic 4

      Fractions

    5. Topic 5

      Percentages

    6. Topic 6

      Simple & Compound Interest, Growth & Decay

    7. Topic 7

      Fractions, Decimals & Percentages

    8. Topic 8

      Rounding, Estimation & Bounds

    9. Topic 9

      Surds

    10. Topic 10

      Using a Calculator

    1. Topic 1

      Introduction to Algebra

    2. Topic 2

      Algebraic Roots & Indices

    3. Topic 3

      Expanding Brackets

    4. Topic 4

      Factorising

    5. Topic 5

      Completing the Square

    6. Topic 6

      Algebraic Fractions

    7. Topic 7

      Rearranging Formulas

    8. Topic 8

      Algebraic Proof

    9. Topic 9

      Linear Equations

    10. Topic 10

      Quadratic Equations

    11. Topic 11

      Simultaneous Equations

    12. Topic 12

      Iteration

    13. Topic 13

      Forming & Solving Equations

    14. Topic 14

      Functions

    15. Topic 15

      Coordinate Geometry

    16. Topic 16

      Linear Graphs

    17. Topic 17

      Graphs of Functions

    18. Topic 18

      Equation of a Circle

    19. Topic 19

      Estimating Gradients & Areas under Graphs

    20. Topic 20

      Real-Life Graphs

    21. Topic 21

      Solving Inequalities

    22. Topic 22

      Graphing Inequalities

    23. Topic 23

      Transformations of Graphs

    24. Topic 24

      Sequences

    1. Topic 1

      Ratios

    2. Topic 2

      Problem Solving with Ratios

    3. Topic 3

      Direct & Inverse Proportion

    4. Topic 4

      Standard & Compound Units

    5. Topic 5

      Exchange Rates & Best Buys

    1. Topic 1

      Symmetry & Shapes

    2. Topic 2

      Angles in Polygons & Parallel Lines

    3. Topic 3

      Bearings, Scale Drawing, Constructions & Loci

    4. Topic 4

      Circle Theorems

    5. Topic 5

      Area & Perimeter

    6. Topic 6

      Circles, Arcs & Sectors

    7. Topic 7

      Volume & Surface Area

    8. Topic 8

      Congruence, Similarity & Geometrical Proof

    9. Topic 9

      Area & Volume of Similar Shapes

    10. Topic 10

      Pythagoras & Trigonometry

    11. Topic 11

      Sine, Cosine Rule & Area of Triangles

    12. Topic 12

      3D Pythagoras & Trigonometry

    13. Topic 13

      Vectors

    14. Topic 14

      Transformations

    1. Topic 1

      Introduction to Probability

    2. Topic 2

      Simple Probability Diagrams

    3. Topic 3

      Tree Diagrams

    4. Topic 4

      Combined & Conditional Probability

    1. Topic 1

      Averages, Ranges & Data

    2. Topic 2

      Statistical Diagrams

    3. Topic 3

      Histograms

    4. Topic 4

      Cumulative Frequency & Box Plots

    5. Topic 5

      Scatter Graphs & Correlation