Human fingerprints are detailed, unique and difficult to alter, making them suitable as long-term markers of human identity. While algebra may not have fingers, they do have their own identities and characteristics as well.
Algebraic identities are “fancy names” for important equations that will help us in simplifying an algebraic equation.
The 3 most important and commonly-used identities are:
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
(a + b) (a – b) = a2 – b2
If this is your first time seeing these 3 equations, don’t be thrown off just yet! These equations might not make any sense to you right now, but they would be very helpful in future. Algebraic identities typically consist of 2 letters (sometimes 3 or more!).
Don’t be afraid of the 2 new letters you see there! They are just letters that represent the ‘constants’ (a value that does not change) in an equation.
Let’s look at the letter a = 2 and b = 3. That doesn’t look like 4 and 9…that because in the equations the letters are so if then a = 2. Same goes for b making it b=3.
(1) It will help you to understand the topic much better if you are familiar with these identities and if you are able to recognise them in different equations.
E.g. 4x2 + 12x + 9
(2) To tell if an identity can be applied to a quadratic equation, first check if both the coefficients (the number before , x and the constant itself) of and the constant can be square-rooted. In this case, the square root of the coefficient of , which is 4, is 2. Meanwhile, square root of the constant, 9, is 3.
Nope…not this root.
(3) To know if an identity can be applied to a quadratic equation, check if the product of the 2 square roots is equals to the half of the coefficient of x. In this case, it is, since 3 x 2 = 6 and 12/2=6
Now, you’ll be able to apply the identity (a + b)2 = a2 + 2ab + b2
When applied, would become
Congratulations! You have successfully used your first identity!
Other identities are used in a similar fashion. To know which identity is the correct / most appropriate one to be used, find the similarity between the equation in the question and the identities. The most similar-looking identity is the one that should be used.
For now, these equations might be challenging to remember. However, as you use these identities more often, you’ll subconsciously commit them to memory even without much effort ~
We hope that you may now identify and apply these algebraic identities better. Be prepared to learn more algebraic identities as you move on to upper secondary! If you are interested to learn other algebraic identities which are just as important, sign up here for our free trial class!