Completing the square is converting a quadratic equation from to , where are constants.
This is a very helpful technique with several uses.
The vertex of a parabola can be found by completing the square in its equation. Consider a parabola whose equation is Completing the square gives
According to the properties of a square, the value of the square minimizes to zero when the variable being squared equals zero, and the value of the square increases as the absolute value of the variable being squared increases. Thus, our whole expression or will have an extreme value when Plugging into the equation gives and now we know the coordinates of the vertex: where is the quadratic's discriminant.
Finding the range of may require quite a bit of effort. However, once completing the square is done, the range can be found in just a matter of seconds. Upon completing the square, we have
We've discussed in the prior section that this has an extreme value when Whether this is a maximum or minimum depends on the sign of We know that (since it's a square). Hence if the extreme value will be a minimum, and the range will be When the function will have a maximum, and its range will be
Find the range of , where is a positive real number.
The variable term in is . Since it is a perfect square, its range is all non-negative real numbers, that is . After adding , which is a constant, the range of becomes .
Note: We can also find the value of at which the minimum occurs. This is when , that is, .
Find the range of , where is a negative real number.
Now the range of is , since is negative. After adding , the range of becomes , and we are done.
Note: Again, at the extreme point, the -coordinate is .
Find the range of .
After completing the square, we get , implying that its range is .
Many times, it becomes difficult to factorize a quadratic expression, especially when the roots are irrational or complex. This is where completing the square helps. Factorization of any quadratic can be done by completing the square followed by using the identity
So given a quadratic equation if we succeed in bringing it to the form then we can factorize it with the difference of two squares identity to obtain both of its roots easily. The procedure is as follows:
Observe that now the equation has come into the form Then we apply the difference of two squares identity, and we're done!
First, we complete the square:
Now using the identity we get
Using the same method as in the above example, we get