okay let's do some fo4 fun and this video is brought to you by brilliant more about them later
we are gonna solve my first quintic equation that's not easy to factor namely x^5+x^4+1=0
this is not easy to factor because one and negative one they are now solution for this
and the truth is this only has one real solution and that's the only one that we're trying to get
right another truth is that unfortunately we do not have a formula to solve a general quintic
equation it has been proven but I do not know if there's a formula for cream cake that only has the
fifth power term the fourth power term and also Nikken center if you do know about such a formula
please leave a comment down below and let us know by anyway though please pause the video and try
this first okay hopefully gets deposited on try it man I think we still have to try to factor
this because we don't have a formula for that unless you have other metal for this okay let's
see how we can do that where we have up quintic maybe we can break it down to a quadratic time
sir cubic as follows so let me just put this down here for you guys first way let's write this as a
quadratic time sir cubic and I will just write yes because this right here is one so I'll put
an x squared and then the next coefficient I do not know so I just put down a X and then
some constant B and the truth is because we're just adding it one right here this right here
can be one wonder if you want but I don't know I will just put on B so that's the quadratic part
and then the cubic pius x plus their power same reason well 1 and then plus the next constant X
square and then another constant and the last the e like this and the truth is B and E they have to
be both passed you want or maybe both negative one I don't know which one anyway though you
can multiply everything out and then just a quick oil future and you end up with a system equation
with a lot of unknowns and a lot of variables no that's not who that this is overkill alright
that's talked about sounds much much better named D problem solving strategy and my favorite one
among all is that when we are doing something hard try to think about something easier try to think
about something that we can actually do try to sing about something that we have done before well
you see that this right here we are missing extra star power x squared X right maybe if everybody is
present it's actually much easier to factor so let me put that down top of the look if I just
put down extra field power class actually force Plus everybody so I will just say X plus 3 power
plus x squared plus X plus 1 let's focus on this first when everybody is present that's actually
much better because we can factor by grouping you see the first three terms we can factor out X to
the third power and then we will be leaving with x squared plus X plus one great and right here I
will just factor out one and we still get the same thing x squared plus X plus one perfect again we
can factor that out so x squared plus X plus one and I will just leave this as holy s which is X
cube plus one I know this is factorable but let me not do that you will see why now this is what we
are going to do we are going to compact Europe we are going to invite these people to their
but we also to take them away so we don't change the value and we can actually factor this nicely
have a look right here let me write yes expletive power plus X to the fourth power but I will have
to add X plus their power x squared and then X and then I will put down the plus one again
this right here have to be subtracted right away so let me just - its to a third power - square
minus X all in all everybody has to be equal to zero all right pretty good well I can make
it better you see everybody here it's exactly what we have here just act it out right away
namely we have X square plus X plus 1 times X to the third power plus one like this and then this
guy's we can factor out a negative X isn't it yes and then we have the rest which is x squared plus
X and then plus one and still everybody would be equal to zero and really product myself like an
actually fitting everything right here all right oh this is so good because this and that are the
strength so that means we can factor that out so I will have x squared plus X plus one and
then I will just have this part left so I will just put this down yes let me just do the same
blue we have X to the third power plus one but we have the minus X let me just put that down here
like that so put this down in black here we go there we go and that's equal to zero and now of
course we can just use the quadratic formula to solve this but you will see this right here does
not give you any real solutions the only real solution is coming from the cubic and this cube
is actually really nice because in the form now we can just use the cubic formula right away or
aka the Cardinal formula aka the calcaneus formula I know the storage behind those two people right
all right so let me put that down right here for you guys notice if we have a cubed in the
following form namely X plus their power plus P X to the first and then plus Q if all this is equal
to 0 then by the cubic formula we will have X being equal to you open a cube root first and
then you can remember qqp and you do the following negative q and you divide it by two and then you
add you open a square root and then sidewise you will have Q square and then remember two squared
is four and then you add P to the third power and remember three to the third power is 27 just like
this we are not done yet this another part but the other part it's actually very easy because we just
have to combine this with pretty much the same thing except work the plus right here it will be
minus that's all so inside here we have negative Q over two minus the square root and here we have
the Q square over 4 and then plus P to the third power over 27 just like that all right so we can
come back here and then you see this right here tells us P use negative 1 and Q years positive 1
so we just have to turn everything to the formula and see if I can fit in everything right here or
not so let's see X is equal to the cube root right first we have naked heat Q over 2 so we
have to make a Q point 1/2 negative 1 over 2 and then we add we open the square root and then we
have 2 squared which is just 1 squared which is just 1 over 4 and then we combine that with P is
negative 1 to the third power which is still negative 1 so this becomes negative and then
we have 1 and this is over 27 all right that's the first part and the second part is which was
change the plus to minus so I will just see we are going to combine the cube root of negative
1/2 and then - and then we have the square root and then we have 1 over 4 minus 1 over 27 just
like that yes everything every single yeah cool and of course if you want to be fancy you can
combine the fractions and then simplify whichever way that you would like but I would just like to
tell you the answer is approximately negative one point three two four seven like this hey if
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