Alright we going to look at how to derive the quadratic formula which say that if we have a generic quadratic equation we put it in this form 
where  a, b and c is the coefficients. 
Then we can say that X is a variable here. 
Then X will be negative b plus or minus the square root of all of b squared minus 4ac and all of that over 2a. 
A couple of things about the quadratic formula make sure that you’re fraction bars are underneath all of these 
and the other thing is to make sure that your square root is over all of these. 
Okay so what will do is we will derive the quadratic formula  by solving this equation here.
And I am going to do this by completing the square. 
The quadratic fomula s just completing the square basically in a formula  sort of form .
So the first thing is divide everything by a so that we have a 1 in front of the X square. 
Now we need to move the constant term C over A to the right side and that becomes negative C over A.
And the interesting thing now is we need to complete the square on the left side.
So add something to the left side to complete the square.
And I have to add the same thing to the right side to keep it balanced. 
And so what I do is I take half of this middle coefficient. 
So one half times B over A. And I square that. 
And so I get B square on top. On bottom I have a 2a. And when I square that I get 4a square. 
So I have B square over 4 a square here. 
Okay? And then on the left side I need to factor it because I completed the square. 
It’s going to be something squared to get the X squared  in in X.
And then my term that goes here is half of the B over A. Half of B over A as we already saw is B over 2 a.
And then on the right side you need a common denominator to add these together, which is 4 a squared. 
So I move my top and bottom over here by 4a.
And so I end up with negative 4ac plus B square all over  4a square. 
Now that I have completed the square I need to take the square root of both sides. 
So I got X plus B over 2a equals plus or minus the square root of the entire fraction. 
I am going to rewrite my numerator as B squared minus 4ac just to make it match what we have up here. 
And then my denominator is 4a square. Make sure it is the square root of the whole thing.
Now I can simplify the right side a little bit. 
I can split my square root . Just take the square of the  top and bottom separately. 
So the square root of the top is the square root of the top and the square root of the bottom 4a square, the square root of that is 2a.
Finally I can subtract my B over 2a over. 
And since these two fractions have the same denominator, I can put them together with the plus or minus on the top.
So again what we’ve just done is proven that this quadratic equation here which a, b and c  could be any numbers 
except  a can’t be zero because if a were zero, it won’t be a quadratic. 
But if you have an equation of this form, of any numbers a, b and c you can plug it into this formula and it will tell you what X is. 
Okay?
So all that the quadratic formula is, is a formula that is derived by completing the square to solve 
the quadratic equation.
