 
Welcome back to speller tutorial services in today's video
We're going to derive the quadratic formula given a quadratic equation in standard form here to the right
We have the quadratic formula negative b plus or minus the square root of b squared minus 4ac all of that divided by 2a
here the quadratic equation a
quadratic equation in standard form ax
squared plus BX plus C is equal to 0 where a B and C are real numbers
We're going to use the completing the square method and to make that conversion easier
We're going to divide each term by a because we want the coefficient of this x squared term to just be one
Alright in doing so these 8's cancel leaving us with an x squared
plus B over a times X
Plus C over a is equal to 0 and we want to subtract the quantity C
Over a from both the left and right hand sides so those cancel
and I'm left with x squared plus B over a times X is equal to negative C over a
Now I want to use the completing the square method and to use the completing the square method
I need to find a quantity to add to both the left and right hand side that
also makes the left hand side a perfect square trinomial and
The method to do that is to take the coefficient of the X term in
this case that coefficient is B is B over a and I
Need to divide that by 2 and
Because this is going to be a complex fraction. I'm making that into 2 over a
So now I need to begin to simplify this expression
First step here is the B over?
it becomes B over 2a and
That expression is squared and that becomes B squared over
4a squared where 2 times 2 or 2 to the second is 4 and
A to the 2nd is a squared so the quantity to add to both the left and right hand side
Left and right hand sides are B squared over
4a squared
All right now this left-hand side here is a perfect square trinomial
so we'll continue here off to the right where this becomes X plus and
The term that of the expression here is B over. 2a that becomes the second term in this binomial
and
This binomial is squared and that there is equal to and I'm gonna switch the order here
Just to make this look more like the equation that I want to derive so this is gonna be B squared
over
4a squared
minus
C over a I notice here that I don't have common denominators, I'm trying to subtract so I need common denominators
to condense this
expression here to the right so that means we're going to multiply the numerator and
denominator by 4a
So let's rewrite the left-hand side X plus
B over 2a that quantity is squared is equal to B squared minus
well C times 4a gives me 4ac and
For a times a and gives me this 4a squared
All right and now to get rid of the squared
I need to take the square root of both sides because I eventually need to solve for that for the
Variable X so I'll take the square root of the left and take the square root of the right and this here becomes X plus
B over 2a is equal to
Plus or minus, and I'm gonna take the square root
and I'm gonna apply it to both the numerator and denominator separately so the numerator is b squared minus 4ac and
The denominator is the square root of?
4a squared notice that the denominator whoops and let's make that look like a squared
whoops
the square root of
4a
squared notice in my denominator that can be simplified the square root of 4 is 2
and the square root of a squared is a
So now let's rewrite one more time
X plus B over, 2a is equal to plus or minus
the square root of b squared minus 4ac
Where all that is over the simplified?
Denominator 2a and now lastly to get this X term by itself
I need to subtract B over 2a from both the left hand side and
From the right hand side in
Doing so those cancel, and I'm left with X is equal to negative B over 2a
plus or minus the square root of b squared minus 4ac and
all that's over 2a and
notice again that and both of these fractions the denominator is 2a, so I can rewrite this as one fraction so
negative B plus or minus the
square root of b squared minus 4ac
where all of that is
Over 2a and notice that this here matches the quadratic formula that I had written up here from the start
Alright, that sums it up for this video by speller tutorial services
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