Today I'm showing how you derive and
prove the quadratic formula but with calculus.
Usually this is done by
completing the square but today I'm
going to show an even more elegant way
with calculus one that you probably never heard about.
The quadratic formula
is a formula that solves for the roots
of any quadratic equation which is in
the form of f(x) = ax^2 + bx +c
We can take the derivative of
this to find f'(x) with the power rule.
We take the 2 down that's 2ax and then we plus b.
Since C is a constant we just ignore that from what we know about integral calculus
f(x) is equal to the integral of f'(x)
because the integral is basically the
antiderivative and we can substitute f'(x) here to be 2ax+b
but we can't take this integral by the
way because let's say if we do take the
this indefinite integral through the
reverse power rule we get (2ax^2)/2 + Bx
Then we have to add
the constant of integration the twos
cancel each other out and we have ax
squared plus BX plus C which is our
original equation and that's not exactly
helpful so we need to do something very
interesting would we turn this
indefinite integral into a definite
integral from 0 to X of 2a T plus B D T
and then since this is a definite
integral there's no constant so we have
to add that at the front
so this is our new f of X equation but
we still can't directly solve this right
away because if we do try to we get an
equation that looks like this
280 squared over 2 plus BT and then we
take it from X from 0 to X plus C at the
end and when T is X we just get ax
square plus BX and when T is 0 that's
just zeros at the end we get ax squared
plus BX plus C all over again so we're
going to we're going to take this
integral and this indefinite integral
but and a slightly strange way we're
going to use u-substitution
we set u to 280 plus B then our D U the
derivative would be 2a the T so remember
2 a is a constant so we have to find a
way to stick to a somewhere into this
definite integral you can do this by
multiplying the in depth that definite
integral by 1 over 2a on the outside
from zero to X of 2 that so we have our
2 a and then we can multiply that by 280
plus B D T DT so now that everything is
set we can finally do our u substitution
to move everything from the key world to
the new world
so here we have C plus 1 over 2 a
definite integral from 0 to X like that
na of
you and since D U is to ADT we just have
you D you but we can't exactly solve for
you yet because we forgot to change the
limits of integration because this is
still taking um the integral from when P
is equal to 0 to when T is equal to X so
when you change the limits of the
integration to the erode as well so we
can do this by plugging them into our u
function which is 280 plus B here so
let's rewrite this as c plus 1 over 2a
of the integral so when u is equal to 0
let's when u is equal to 0 we have 2a 0
plus B which is just equal to B and then
when u is equal to X we have 2a ax plus
B so we can take it to ax plus B right
there you D u so now that everything is
converted we can finally take this
integral by using the reverse power rule
remember we have C plus 1 over 2a here
and then we have u squared over 2 from
b2 to ax plus B so now we just have to
substitute 2x plus B into you and
subtract that by substituting B into u
squared so let's expand this C plus 1
over 2a and it's going to be a monster
equation that's going to be 2x
plus B squared over 2 minus B squared
over 2 and then remember this is still f
of X we just made it a whole lot
complicated but don't worry from here
it's going to all boil down to the
quadratic formula just watch a key thing
to note here is that youssef odd the
quadric quadratic formula solves for X
when y is equal to 0 so let's replace f
of X with 0 so now we have to all we
have to do is solve for X and I highly
encourage you to pause this video right
now and try it out for yourself so let's
simplify everything C plus 2a X plus B
squared minus B squared and then 2a
times 2 equals 4a equals 0 so now we can
start rearranging everything and just to
make things simpler we're just going to
put the 0 here because I like it this
way we can subtract C from both sides
and multiply it by 4a this will give us
2 ax plus B squared minus B squared
equals don't forget
so basically we can take out the sea
take out the 4a and then move them to
the other side so it becomes negative
for a sea now we can add B squared to
the other side and that will make the
equation to ax plus B squared I think
you can tell where it's going now B
squared minus 4 ay C now we take the
square root of both sides and this will
give us to a X plus B equals the square
root of b squared minus 4 ay C now we
can subtract B from both sides and also
divide both sides by 2a and we get this
neat equation of X equals this square
root of b squared minus 4ac minus b over
2a and remember that the square root is
always plus or minus and all we have to
do is simple rearrangement to make this
into the formula become and love today x
equals minus b squared minus 4ac over 2a
isn't this beautiful
I had a big square rounded to show that
this is our final answer
and this ladies and gentlemen is how you
prove the quadratic formula with
calculus pretty neat huh
