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Solve equations of the form (x+k)^{2}=d, where k and
d are numbers.
Find a way of figuring out what number to add to both sides
of a quadratic equations so that the left side will become a perfect
square.
To work our way up to the task of solving equations of the form
(x+k)^{2}=d
let's first start with the slightly easier task of solving equation of
the form
x2=d
How do we solve an equation of the form
x2=d?
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
If x less than 0 then what happens? What kind of number can you square and
get a negative number? If you square a positive number then clearly you
get a positive number. But if you square a negative number then you have
a product of two negative numbers, so you still get a positive number. So
what is left for squaring and getting a negative number? Nothing. So the
equation has no solutions.
Now let's look at the more general equation of the form
(x+k)^{2}=d
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 1
Problem: Solve the equation.
Solution:
Example 2
Problem: Solve the equation.
Solution: Anything you can do with x, you can do with x+1, and once you find
x+1 all you have to do to get x is subtract 1.
Completing the Square
Now 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
(x+k)^{2}
and see what makes perfect squares tick. Applying our formula for squaring
a binomial, we get
(x+k)^{2}=x^{2}+2kx+k^{2}
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 k^{2}. 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 k^{2}.
So if you have an expression of the form
x^{2}+bx
and you want to find something to add to it to make it a perfect square,
then you need to
Divide b by 2 to get k
Square k to get k^{2}.
Example
Problem: Complete the square.
Solution:
The yellow part is the scratch paper. On the scratch paper you first divide
the 9 by 2 and then square the result. Don't worry about the minus sign, because
it will go away when you square anyway. Then the number you get will be the
number you need to add to the expression to make a perfect square out of it.
After you do that it is good practice to write it as the square that it is.
For that you can use the first line of your scratch paper and match the sign
with the sign of the second term of the original expression.
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 Square
Now we are ready to use the method of completing the squares to solve quadratic
equations. The best way to do this is as follows.
Add something to both sides so that the left side has no constant
term.
Figure out what to add to the left side to make it a perfect
square, and add that to both sides.
Write the left side as the perfect square that it is and do
the arithmetic on the right side.
Solve the equation you get by the methods of equations of the
form (x+k)^{2}=d