If you are a student of algebra, you might have come across the concept of factorising a quadratic expression.
Factorising is an essential skill in algebra, and it involves breaking down an algebraic expression into simpler terms.
One of the most common methods of factorising quadratic expressions is the technique known as the difference of two squares method .
The difference of two squares method is a technique used to factorise quadratic expressions that are in the following format:
a^{2} - b^{2}
This format for this method is where there are two terms, either may contain algebraic variables, and both terms are separated by a minus sign.
Once the format has been identified factorising the expression can be factored into the following expression:
(a + b) (a - b)
Factoring using the difference of two squares method is an essential skill in algebra, and it can help you simplify complex expressions and solve equations.
How to Identify Difference in Two Squares
If you’re studying algebra, then you’ve probably come across the phrase “ difference of two squares .”
This is a specific type of binomial expression that can be factored quickly and easily provided you correctly identify the expression .
The key to factoring this type of expression is to recognise that the two terms are squared terms .
Therefore, it helps to remember what square numerical coefficients are. Here’s a reminder of the first ten squares:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100...
If there is a variable in either term, these also need to be square variables, such as x^2, x^{2}y^{2} .
Therefore, the following expressions can use this method:
- 1 - x^2
- 4 - x^{2}y^{2}
- 100x^{2} - 25y^{2}
It’s important to note that this method only works when the two terms in the binomial are perfect squares, and they are being subtracted from each other.
If you have a binomial expression that is being added or multiplied, or if the terms are not perfect squares, then this formula won’t work.
Instead of using trial and error or other methods, you can apply this method to get the answer quickly and easily.
Once you have correctly identified a binomial expression that fits the above pattern, you can proceed to factorising the expression into the format:
(a + b) (a - b)
Simple Checklist
Here’s a quick checklist you can use to determine if you can apply this method to an expression:
-
Are there two terms?
- Yes (continue on)
- No (cannot use this method)
-
Can the two terms be arranged such that one is being subtracted from the other?
- Yes (continue on)
- No (cannot use this method)
-
Is the first term a square?
- Yes (continue on)
- No (cannot use this method)
-
Is the second term a square?
- Yes (continue on)
- No (cannot use this method)
- If you’ve arrived here, you can safely apply this technique a^{2} - b^{2} = (a + b)(a - b) .
Examples
Here are some examples demonstrating how to factorise the difference between two squares:
Example 1
x^{2} - 16
Using the above checklist here’s how the above would look:
- Two terms? Yes.
- Two terms being subtracted? Yes.
- Is the first term x^{2} a square? x^{2} is the same as x \times x in expanded form, and you know that a square is the product of two same expressions – which x^{2} is. You can learn more about square numbers here . Therefore this is a Yes.
- Is the second term 16 a square? Yes. 16 is a square number.
As you’ve satisfied all the conditions to apply this method, the above expression can be factorised, therefore the answer would be:
(x + 4)(x - 4)
Example 2
9y^{2} - 64
Once again, using the above checklist would be as follows:
- Two terms? Yes.
- Subtraction between them? Yes.
- Is the first term 9y^{2} a square? It is. 9y^{2} is the product of 3y \times 3y .
- Is the second term 64 a square? It is. 64 is the product of 8 \times 8 .
Therefore, this method can be applied and the factorisation of this expression would be:
(3y + 8)(3y - 8)
Example 3
25 - x^{2}
- Two terms? Yes.
- Subtraction? Yes.
- First term 25 a square? Yes.
- Second term x^{2} a square? Yes.
Then, the expression can be written as:
(5 + x)(5 - x)
These are just a few examples of how to factorise using the difference of two squares method.
It is important to note that not all quadratic expressions can be factored using the difference of two squares method . A quadratic expression can only be factored using this method if it is in the form of a^2- b^2 .
Additionally, it is important to simplify the expression before attempting to factor it using the difference in two squares method. This involves looking for common factors that can be factored out of the expression.
In summary, identifying a difference in two squares involves looking for two perfect squares that are being subtracted from each other. Not all quadratic expressions can be factored using this method, and it is important to simplify the expression before attempting to factor it using this method.
Conclusion
By mastering the technique of factoring the difference of two squares, you can simplify complex algebraic expressions and solve problems much more efficiently. Remember that the difference of two squares pattern only applies when you have two perfect squares with a minus sign between them.
When factoring a quadratic expression, always try to factor out the greatest common factor first. If the expression is not a perfect square trinomial, check if it can be factored using the difference of two squares pattern. If it cannot be factored in this way, try other factoring techniques such as grouping or the quadratic formula.
It is important to practice factoring exercises regularly to improve your skills and speed. As you become more familiar with the process, you will be able to factor expressions more quickly and accurately.
Remember, factoring is an essential skill in algebra and is used in many areas of mathematics and science. By mastering factoring techniques, you will be better equipped to tackle more advanced problems and succeed in your studies.