What do you think a formula is? Think about it. Do you think it is something that has an equal to sign in it? Are all = sign mathematical items formulae? No?
Now here’s what a formula’s definition is: a mathematical relationship or rule expressed in symbols. This is just what it is using complex words. To simplify, it is a uniform way of finding an answer. So basically using a formula you can find the answer to anything, crazy isn’t it?
We need and use these formulae to make life simple for us, instead of we doing the whole process every time. We can just input certain things into specific formulae and get our answers. We use formulae everywhere in our life, we just don’t realize it. Our brain functions on formulae and logic. Only that way we carry out our actions to certain stimulations. So there you go, now you know we use formulae in everything. Math is famously known for its formulae simply because it has so many of them. Although, do you know how we got to them?
Now that’s a really important question, how do we get or make formulae?
Pretty tough isn’t it?
It takes decades and centuries of research and multiple extraordinary mathematicians for just one formula to be formed. So, that’s how difficult it is just to make one formula. Now think about it we have so many formulae. How much time and effort must’ve been put into it! Although do you ever wonder how they got to the formulae? What must they have done? That’s the difficult part and that’s why we derive formulae to get to the root of it and to understand it, cause isn’t it better to understand the formula properly and know how it works from the bottom rather than just by hearting it. So, that’s why we derive formulae and we should always understand them properly. For instance, here I am going to derive the quadratic formula for you:
What is the quadratic formula?
Why do we use it? To find a solution of x, but from what? The quadratic equation.
What is the quadratic equation?
Now here let’s analyse the equation. There’s a common variable ‘x’ which we have to find and then there’s ‘a’, ‘b’ and ‘c’ as letters. Now what are these letters? These are nothing but the coefficients, otherwise known as numbers. We have used these letters a, b and c as we do not know these coefficient values and that’s the beauty of formulae. I can add any numbers instead of a, b and c and get my answer. All of this equates to 0 which makes it simple.
Next, the one thing you need to know is that the whole quadratic equation is centered and found by completing the square basically. So, what we now do, is complete the square and only then can we derive the formula as the formula is based on this.
First, to complete the square, the rule is that the coefficient to the squared term which is x2 should be 1. Now look up to the equation is it 1? No, it’s a, so now to remove this ‘a’ from there we have to divide the whole thing by ‘a’ such that we can nullify the ‘a’ on the x2 term. This is because as ‘a’ is in multiplication, we have to use division to remove it. So now the formula looks something like this:
Here there is no ‘a’ on x^2 as it is cancelled from both the numerator and denominator. And 0 divided by a will be 0.
Now we have to complete the square for ‘x’ so which all terms have ‘x’ in them? That’s right the first and second. We don’t need the ‘c’ term on the left hand side (LHS) so let’s move it to the right hand side (RHS) so that it looks like this:
This c term becomes negative because it was positive on the LHS so to bring it to the RHS, we have to switch signs.
Moving on, now comes the interesting part, the most important part: the part where you actually complete the square. Here, we have to complete the square of LHS. For that, there is a simple rule we can follow which is we half and square the term which has the variable with the power of 1 which would be this term:
Although while finding the last term, we do not have to use the variable as it is the first term-
Completing the square steps:
So now we half it and then we square it
1)Half it: (b/a)/2) so that would basically be (b/2a)
2)Square that now: (b/2a)^2
3) Add this to the LHS
Since we have added this new term, we cannot just leave it like that otherwise this whole thing will become imbalanced and will not work. So to balance it out we add the same thing on the RHS to nullify these. The whole equation looks like this:
Now observe the LHS carefully, do you see something cause I do. It’s a perfect square! That’s why we added the last term. And now since it’s a perfect square, we can simplify even further as we already know the identity. So as this is in an expanded form, we can compress it using identities.
That’s on the LHS, for the RHS we can easily expand the bracket and square that term.
So all in all this would look like:
Here we have simplified the LHS in the identity of (a+b)^2 = a^2 + 2ab + b^2. So we have simplified it backward.
Although why haven’t we taken (a-b)^2? This is because there is no negative sign in the LHS hence we use the first identity itself.
Now the RHS has two separate fractions which do not look that neat so how about we simplify them and make both they’re bases the same. For that we use LCM. As the c/a already has an a in the denominator, we can multiply that fraction by 4a such that both the denominators become 4a^2, although now even c will be multiplied with 4a. So this is how it’ll look:
Still looks a little messy right. But don’t worry we have completed stage one and we’re getting closer.
Next, since we have completed the square, now our aim is make x the subject as that is what the solution is.
The first step towards this is removing the square or the index and that is done by square rooting both the sides (LHS and RHS). here’s how it goes:
Now this is how the equation looks, although we can further simplify the RHS’s denominator so let’s do that so that the root is only of the numerators. This is how the RHS looks like:
Now it’s looking somewhat like it isnt it that’s cause we’re very close. Now we have to make x the subject hence we have to make the LHS only with x and move everything else to the RHS and since the 2 fractions have the same denominator, we can simply bring them together under one fraction to form what? You guessed it! The quadratic formula:
There you go! You have derived the quadratic formula. It wasn’t difficult right. And now you understand this better as well.
These quadratics have so many applications and uses and are so important to us as we need to use it in so many places. So now since you have a strong understanding of it, you can use this properly and be right about it instead of guessing it and hoping you are correct
This is just one of many formulae to exist, but now since you derived this, it is much better right? Now you should always try to derive and understand the why behind things and get to know where it all starts. Only then you can use the formulae properly. This shouldn’t only be for formulae but in fact for anything you learn instead of just by hearting it and remembering it for a couple weeks, understand the concept and remember it for the rest of your life.