Completing the Square (OCR GCSE Maths)
Revision Note
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MarkExpertise
Maths
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
Exam Tip
- 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 .
Find half of +6 (call this p)
Write x2 + 6x in the form (x + p)2 - p2
is the same as
Put this result into the equation of the curve
Simplify the numbers
Use that the turning point of is at
p = 3 and q = -20
turning point at (-3, -20)
(b)
Write in the form
Factorise -2 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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