Hey guys, today I'm going to talk about
how to switch between general form and
turning point form for a quadratic
equation so why would you want to switch
between two ways of expressing the same
equation well let's have a look at this
equation here y equals x squared plus 6x
plus 14 if we try to graph that equation
let me just get rid of my previous graph
so if we try to graph y equals x squared
plus 6x plus 14 we get this graph here
and let me just zoom out a little bit so
we can see that the turning point is
located at x equals negative 3 and y
equals 5 now but however this equation
doesn't tell us anything about the
turning point it's not until we draw the
graph we can see where the turning point
of this graph is located now however if
you have a look at this other equation y
equals x plus 3 in brackets squared plus
5 if we if I try to draw that graph so
I'm going to delete this graph y equals
bracket X plus 3 bracket squared plus 5
you get the exact same graph now however
in this equation we can see already that
the turning point is x equals negative 3
and y equals 5 and that tells us as part
of the formula so when an equation when
a quadratic equation is in turningpoint
form we can already see that the turning
point exists so when y equals
X minus B squared
sorry let me just change that so when x
equals x plus a squared plus B we know
that the turning point will be at x
equals negative a and y equals b right
we can see that in our graph so this is
a turning point form a is 3 so the
turning point here exists at x equals
negative a which is negative 3 and the
turning point exists at y is equal to 5
so b is equal to 5 which means again
Y is equal to 5 for our turning points
for the graph so the question now is how
do I switch from this form into this
form because once I switch once I
rewrite the equation into this new
turning form turning point form I can
see where the turning point is without
even drawing the graph so the process
from between the process to switch from
our general form to our turning point
form is called complete the square
it's called complete the square
because we have a bracket squared and so
therefore this process is called
complete the square
and I'm going to teach you how to do
that using this example so let's firstly
have a look at our perfect square
formula which says that a plus b squared
is equal to if you expand it out it will
give you a squared plus 2 ab plus b
squared right if you're not sure how to
expand brackets have a look at the
previous expanding tutorial so as you
can see here we want to somehow
manipulate our equations so that we can
put these three terms back into
something squared plus a constant term
so let's just have a look at the first
two parts here of this equation so this
first two terms of our equation matches
our first two terms in our perfect
square formula where we have x squared
plus 6x all right we just don't know
what we should put in here and as you
can see in our formula a is X so we know
that it'll be X plus we don't know what
b is squared
so what are we going to put here for b
squared and what are we going to put in
this box for b now if we know that 2ab
is equal to 6x which we can see matching
here 2ab is 6x and we know that a is
equal to X so we can deduce what B is
equal to so 2 times X times B is 6X I
now the B and a 6 look a little bit
similar so we can rearrange this
equation to make B on its own so B is
equal to 6x divided by 2x which is equal
to 3
because 6 divided by 2 is 3 over 1 so B
is equal to 3 so we can simply now put
in this box 3 X plus 3 squared is x
squared plus 6 X plus 3 squared which is
x squared plus 6x plus 9 now notice how
in the equation that we try to convert
it has x squared plus 6 X plus 14 so
basically what we're going to have to do
now is so we have the equation x squared
plus 6x plus 14 and we want to complete
the square
on this equation so what we're going to
do is make it equal to zero and we're
going to move to fourteen to the other
side so I have x squared plus 6x equals
negative 14 and now because we know that
we can put these back into brackets if
we had x squared plus 6x plus 9 so what
I'm going to do now is plus 9 on both
sides so I'm going to write plus 9 on
both sides notice how this equation has
not changed at all right because you can
do whatever you want as long as you do
the same thing to both sides so
therefore we on the left hand side we
now have X plus 3 squared equals
negative 14 plus 9 which is negative 5
and now we can simply move the minus
five to the other side which makes that
X plus three squared plus five is equal
to zero
so we have just completed the square of
our equation x squared plus 6x plus 14
is equal to X plus three squared plus
five which is that what we wanted to do
in the beginning changing from this form
into this form changing from the general
form to the turning point form so now we
know without even drawing the graph that
this equation y equals x squared plus 6x
plus 14 will have a turning point at X
is equal to negative 3 and Y is equal to
5 okay that concludes our tutorial on
how to change from general form and
turning point form thanks for watching
see you next time.
