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To work our way up to the task of solving equations of the form
let's first start with the slightly easier task of solving equation of the form
How do we solve an equation of the form
If x is greater than 0 then the obvious answer is
but this is not quite right because it only gives you the positive square root of d, and all positive numbers have two square roots, a positive one and a negative one. So to be sure that you are getting all solutions to an equation of this form, your answer must be
Now let's look at the more general equation of the form
This is really not much harder since anything you can do with x you should be able to do with x+k. x+k represents a number too. So solve for x+k and then add something to both sides of the equation to get x alone.
Example 1Problem: Solve the equation.
Example 2Problem: Solve the equation.
Completing the SquareNow to problem number two, that of finding something to add to a quadratic to make it a perfect square.
This is what is meant by completing the square, and the secret to it is to expand out the expression
and see what makes perfect squares tick. Applying our formula for squaring a binomial, we get
The key here is to look at the relationship between the coefficient on x and the constant coefficient. The coefficient on x is 2k and the constant term is k2. This means that if we know the coefficient on x, and we want to know what the constant term has to be for the expression to be a perfect square, then we need to divide the coefficient on x by 2 to get k, and then square to get k2.
So if you have an expression of the form
and you want to find something to add to it to make it a perfect square, then you need to
ExampleProblem: Complete the square.
I hope the above has helped you understand the process of completing the square. If not, there is another approach to it that I have written an article about that you might find interesting for further understanding. It is a geometrical approach based on the method that many earlier mathematician used. You can read my article A Geometrical Approach to Completing the Square to find out about it.
Solving by Completing the SquareNow we are ready to use the method of completing the squares to solve quadratic equations. The best way to do this is as follows.