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
- 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:
- Is the same as ?
- This works for negative values of too
- can be written as which is
- A negative 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
- Use the rule above to replace the first two terms, , with
- add 9:
- simplify the numbers:
- answer:
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 and terms only
-
- Use square-shaped brackets here to avoid confusion with round brackets later
- For example,
-
- 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 outside the square brackets and add the
- This looks far more complicated than it is in practice!
- Usually you are asked to give your final answer in the form
- Here
- For quadratics like , do the above with
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 -coordinate
- This links to transformations of graphs (translating by to the left and up)
- If then the turning point is still at
- It's a minimum point if
- It's a maximum point if
- 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
- Expand your answer to check that you have completed the square correctly.
Worked example
Find half of (call this )
Factorise out of the first two terms only
Use square-shaped brackets
Complete the square on the inside the brackets (write in the form where is half of )
Simplify the numbers inside the brackets
is
Multiply all the terms inside the square-shaped brackets by
Simplify the numbers
This is now in the form where , and