GCSEFurther MathsAQARevision Notes2. Algebra & FunctionsAlgebra ToolkitCompleting the Square

Completing the Square (AQA GCSE Further Maths)

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Completing the Square

How can I rewrite the first two terms of a quadratic expression as the difference of two squares?

  • Look at the quadratic expression x2 + bx + c 

  • The first two terms can be written as the difference of two squares using the following rule

x squared plus b x is the same as open parentheses x plus p close parentheses squared minus p squared where p is half of b

  • Check this is true by expanding the right-hand side

    • Is x squared plus 2 x the same as open parentheses x plus 1 close parentheses squared minus 1 squared?

      • Yes: (x + 1)(x + 1) - 12 = x2 + 2x + 1 - 1 = x2 + 2x

  • This works for negative values of b too

    •  x squared minus 20 x can be written as open parentheses x minus 10 close parentheses squared minus open parentheses negative 10 close parentheses squared which is open parentheses x minus 10 close parentheses squared minus 100

    • A negative b does not change the sign at the end

How do I complete the square?

  • Completing the square is a way to rewrite a quadratic expression in a form containing a squared-bracket

  • To complete the square on x2 + 10x + 9

    • Use the rule above to replace the first two terms, x2 + 10x, with (x + 5)2 - 52

    • Add 9:  (x + 5)2 - 52 + 9

    • Simplify the numbers:  (x + 5)2 - 25 + 9

    • Answer: (x + 5)2 - 16 

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

  • You first need to take a out as a factor of the x2 and x terms only

    • 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

      • Use square-shaped brackets here to avoid confusion with curly brackets later

  • Then complete the square on the bit inside the square-brackets: x squared plus b over a x

    • This gives a open square brackets open parentheses x plus p close parentheses squared minus p squared close square brackets plus c

      • where p is half of b over a

  • Finally multiply this expression by the a outside the square-brackets and add the c

    • a open parentheses x plus p close parentheses squared minus a p squared plus c

    • This looks far more complicated than it is in practice!

      • Usually you are asked to give your final answer in the form  a open parentheses x plus p close parentheses squared plus q 

  • For quadratics like negative x squared plus b x plus c, do the above with a = -1

Completing the square Notes Diagram 2, A Level & AS Level Pure Maths Revision Notes

How do I find the turning point by completing the square?

  • Completing the square helps us find the turning point on a quadratic graph

    • If y equals open parentheses x plus p close parentheses squared plus q then the turning point is at open parentheses negative p comma q close parentheses

      • Notice the negative sign in the x-coordinate

      • This links to transformations of graphs (translating y equals x squared by p to the left and q up)

    • If y equals a open parentheses x plus p close parentheses squared plus q then the turning point is still at open parentheses negative p comma q close parentheses

      • It's at a minimum point if a > 0

      • It's at a maximum point if a < 0

  • It can also help you create the equation of a quadratic when given the turning point

Completing the square Notes Diagram 3, A Level & AS Level Pure Maths Revision Notes

  • It can also be used to prove and/or show results using the fact that any "squared term", i.e. the bracket (x ± p)2, will always be greater than or equal to 0

    • You cannot square a number and get a negative value

Completing the square Notes Diagram 4, A Level & AS Level Pure Maths Revision Notes

Examiner Tips and Tricks

  • To know if you have completed the square correctly, expand your answer to check.

Worked Example

(a) By completing the square, find the coordinates of the turning point on the graph of y equals x squared plus 6 x minus 11.

Find half of +6 (call this p) 

p equals 6 over 2 equals 3 

Write x2 + 6x in the form (x + p)2 - p2

x squared plus 6 x is the same as open parentheses x plus 3 close parentheses squared minus 3 squared 

Put this result into the equation of the curve

y equals open parentheses x plus 3 close parentheses squared minus 3 squared minus 11 

Simplify the number

y equals open parentheses x plus 3 close parentheses squared minus 20 

Use that the turning point of y equals open parentheses x plus p close parentheses squared plus q is at open parentheses negative p comma q close parentheses 

p = 3 and q = -20

Turning point at (-3, -20)

 

(b) Write negative 3 x squared plus 12 x plus 24 in the form a open parentheses x plus p close parentheses squared plus q
 

Factorise -2 out of the first two terms only
Use square-shaped brackets 

negative 3 open square brackets x squared minus 4 x close square brackets plus 24 

Complete the square on the x2 - 4x  inside the brackets (write in the form (x + p)2 - p2 where p is half of -4

negative 3 open square brackets open parentheses x minus 2 close parentheses squared minus open parentheses negative 2 close parentheses squared close square brackets plus 24 

Simplify the numbers inside the brackets
(-2)2 is 4

negative 3 open square brackets open parentheses x minus 2 close parentheses squared minus 4 close square brackets plus 24 

Multiply -3 by all the terms inside the square-shaped bracket

negative 3 open parentheses x minus 2 close parentheses squared plus 12 plus 24 

Simplify the numbers

negative 3 open parentheses x minus 2 close parentheses squared plus 36 

This is now in the form a(x + p)2 + q  where a = -3, p = -2 and q = 36

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