Completing the Square
Neil Trivedi
Teacher
Contents
Completing the Square
In maths, we use completing the square for several reasons. When solving equations, we usually try to rearrange to make the unknown the subject. With quadratics, this is more difficult because appears in more than one term.
So far, we have used factorising to solve quadratic equations. However, not all quadratics can be factorised easily, or at all. This is where completing the square becomes useful.
It allows us to rewrite a quadratic so that the unknown is grouped into one place, which makes it easier to solve without factorising.
Completing the square is also useful for sketching quadratic graphs. You can find out more in our Sketching Quadratics note.
When completing the square, we aim to rewrite a quadratic of the form
in the form
The expression is a perfect square, which is where the name “completing the square” comes from. In this form, the appears in one place, inside a single squared bracket.
Completing the square is based on reversing the expansion of a perfect square. For example,
Notice that the coefficient of is which is twice the number in the bracket, and that the constant term,is the square of that number,
In general, this pattern always applies:
This means that the coefficient of is always twice the number in the bracket, and the constant term is always the square of that number.
Now, consider the expression This does not include the constant term so it’s not a perfect square.
If we rewrite it as we would be adding an extra which changes the value of the expression.
To correct this, we subtract
This keeps the value of the expression the same, but now it is written using a perfect square.
Completing the square works by rewriting the and terms of a quadratic in this form. Any constant term in the original quadratic expression is then added at the end.
Note: Watch the video provided below where NeilDoesMaths walks through this same idea using shapes before completing extra questions.
So, in summary, when completing the square, we’re following these steps:
1) Halve the coefficient of and write this number in the bracket.
2) Subtract the square of that number.
3) Add any constant term from the original quadratic at the end.
Completing the Square with quadratics where the coefficient of 𝑥² is 1
Example 1:
Express the following in the form where and are constants, by completing the square.
a)
Single Step: Halve the coefficient of write it in a squared bracket, then subtract its square.
Here, the coefficient of is We halve that to get and write that in a squared bracket.
Then, we subtract the square of which will be So, we get
b)
Single Step: Halve the coefficient of write it in a squared bracket, then subtract its square. Then, add the constant term at the end.
Here, the coefficient of is We halve that to get and write that in a squared bracket.
Then, we subtract the square of which will be
The constant term, does not affect the square, so it is included at the end.
c)
Single Step: Halve the coefficient of write it in a squared bracket, then subtract its square. Then, add the constant term at the end.
Here, the coefficient of is We halve that to get and write that in a squared bracket.
Then, we subtract the square of which will be
We then add the at the end.
To simplify, we write as so that we can combine the constant terms into one fraction.
Note: but it is better to work with fractions because squaring fractions is easier than squaring decimals, especially if you do not have a calculator.
Completing the Square with quadratics where the coefficient of 𝑥² is not 1
So far, we have considered examples where the coefficient of is which allows us to complete the square directly.
However, when the coefficient of is not we must first make it because in completed square form, there is no coefficient in front of inside the bracket.
To achieve this, we factorise out the coefficient of from the and terms only.
We then complete the square, expand the first set of brackets and then simplify the expression at the end.
Example 2:
Express the following in the form where and are constants, by completing the square.
a)
Step 1: Factorise out the coefficient of from the and terms.
When we factorise we divide both terms by
Note: We are ignoring the here because in the process of completing the square, the constant at the end is just brought down with no manipulation.
Step 2: For the expression inside the square brackets, halve the coefficient of write it in a squared bracket, then subtract its square.
For the expression the coefficient of is We halve that to get and write that in a squared bracket.
Then, we subtract the square of which will be
Therefore,
Step 3: Simplify the expression by expanding and then adding the constant at the end.
We expand by multiplying each term inside the square brackets by
To note, we need to be careful not to multiply the with as it is protected by the curved brackets.
Once expanded, we get
b)
Firstly, this expression can be written as
Step 1: Factorise out the coefficient of from the and terms.
When we factorise we divide both terms by
Step 2: For the expression inside the square brackets, halve the coefficient of write it in a squared bracket, then subtract its square.
For the expression the coefficient of is We halve that to get and write that in a squared bracket.
Then, we subtract the square of which will be
Therefore,
Step 3: Simplify the expression by expanding and then adding the constant at the end.
We expand by multiplying each term inside the square brackets by and we get
Generally, we write the positive term first so this can be rewritten as
c)
Step 1: Factorise out the coefficient of from the and terms.
When we factorise we divide both terms by
Step 2: For the expression inside the square brackets, halve the coefficient of write it in a squared bracket, then subtract its square.
For the expression the coefficient of is We halve that to get and write that in a squared bracket.
Then, we subtract the square of which will be
Therefore,
Step 3: Simplify the expression by expanding.
We multiply each term inside the square brackets by To note,
So, when expanded, we get
Solving Quadratics by Completing the Square
Example 3:
Solve the following by completing the square:
a)
Step 1: Halve the coefficient of write it in a squared bracket, then subtract its square. Then, add the constant term at the end.
This quadratic cannot be factorised which is why we are choosing to solve by Completing the Square.
Here, the coefficient of is We halve that to get and write that in a squared bracket.
Then, we subtract the square of which will be
We then add the at the end and we get:
Step 2: Solve for
To solve for we need to rearrange the equation to isolate
Adding to both sides,
Square rooting both sides,
Subtracting from both sides,
b)
Step 1: Halve the coefficient of write it a squared bracket, then subtract its square. Then, add the constant term at the end.
Here, the coefficient of is We halve that to get and write that in a squared bracket.
Then, we subtract the square of which will be
We then add the at the end and we get
To simplify, we write as so that we can combine the constant terms into one fraction.
Step 2: Solve for
We need to rearrange the following equation to isolate
Adding to both sides,
Square rooting both sides,
Just like how we squared the numerator and denominator when working out earlier, the same applies to surds where we can root the numerator and denominator. So,
and we can find that simplifies to give
Subtracting from both sides,
Both fractions on the right side have the same denominator of so we can combine the fractions by adding the numerators to give
c)
Step 1: Factorise out the coefficient of from the and terms.
When we factorise we divide both terms by
Step 2: For the expression inside the square brackets, halve the coefficient of write it in a squared bracket, then subtract its square.
For the expression the coefficient of is We halve that to get and write that in a squared bracket.
Then, we subtract the square of which will be
Therefore, we get
Step 3: Simplify the left side of the equation by expanding and then adding the constant at the end.
We multiply each term inside the square brackets by To note,
So, when expanded, we get
To simplify, we write as so that we can combine the constant terms into one fraction.
Step 4: Solve for
We need to rearrange the following equation to isolate
Adding to both sides,
Dividing both sides by
Square rooting both sides,
We can rewrite the right side by rooting both the numerator and the denominator.
Since
Adding to both sides,
Both fractions on the right side have the same denominator so we can add the numerators to give
To note, at the beginning, we could have first divided our equation through by and completed the square with and ended up with the same solutions. You can only do this with equations not expressions.
Challenging Questions