Completing the square
Completing the square is a method used to rewrite a quadratic expression, from
the form
Completing the square when (a=1)
Section titled “Completing the square when (a=1)”When we have a nice, simple quadratic, in the form
This makes a lot more sense with an example.
Example: Complete the square for
Section titled “Example: Complete the square for ”- Find what
and are: - Calculate
: - If we take
and expand it, we get: - This is what we want, we just have an extra
, so take that away from the expression:
Answer:
Completing the square when
Section titled “Completing the square when ”When the coefficient of
We have an equation equal to zero, which means we can divide by any number on the left side without worrying about changing the equation (as zero divided by any number is still zero).
Our first step is to get rid of the
We can now simply use the method above to complete the square - as we don’t need
to worry about the
Example: Complete the square for
Section titled “Example: Complete the square for ”- First, divide the whole equation by
to get rid of the coefficient: - Now, we can complete the square using the method above:
- So, we need to subtract
from the expression:
- Finally, we need to remember that we divided the whole equation by
at the start, so we need to multiply the whole completed square expression by to get back to the original equation:
Answer:
Finding the turning point from the completed square form
Section titled “Finding the turning point from the completed square form”The turning point of
Notice that we completely ignore the
coefficient when finding the turning point. It’s not important here!
The number in the brackets tells us the
We negate the number inside the brackets to find our