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Syllabus Edition
First teaching 2021
Last exams 2024
Completing the Square (CIE IGCSE Maths: Extended)
Revision Note
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
is the same as where is half of
- Check this is true by expanding the right-hand side
- Is the same as ?
- Yes: (x + 1)(x + 1) - 12 = x2 + 2x + 1 - 1 = x2 + 2x
- Is the same as ?
- This works for negative values of b too
- can be written as which is
- 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 out as a factor of the x2 and x terms only
-
- Use square-shaped brackets here to avoid confusion with curly brackets later
-
- Then complete the square on the bit inside the square-brackets:
- This gives
- where p is half of
- This gives
- Finally multiply this expression by the a outside the square-brackets and add the c
- This looks far more complicated than it is in practice!
- Usually you are asked to give your final answer in the form
- For quadratics like , do the above with a = -1
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 then the turning point is at
- Notice the negative sign in the x-coordinate
- This links to transformations of graphs (translating by p to the left and q up)
- If then the turning point is still at
- It's at a minimum point if a > 0
- It's at a maximum point if a < 0
- If then the turning point is at
- It can also help you create the equation of a quadratic when given the turning point
- 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
Examiner Tip
- To know if you have completed the square correctly, expand your answer to check.
Worked example
Find half of +6 (call this p)
Factorise -3 out of the first two terms only
Use square-shaped brackets
Complete the square on the x2 - 4x inside the brackets (write in the form (x + p)2 - p2 where p is half of -4)
Simplify the numbers inside the brackets
(-2)2 is 4
Multiply -3 by all the terms inside the square-shaped brackets
Simplify the numbers
This is now in the form a(x + p)2 + q where a = -3, p = -2 and q = 36
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