.
.
In this segment we're going to derive the quadratic solution,
so, for the quadratic formula . . . quadratic equation.|So we have a quadratic equation
somebody gives this to us, a general one, and we want to derive the
solution.|So you've already
seen that if you have a x squared plus b x plus c equal to 0,
we already know that the solution looks like this, this quite popular
formula.|So that is the solution to the quadratic equation, and what we're going to do in this
segment, we're going to derive this particular formula, that how does it turn out to be that the solution
is as follows.|Now, when you have a x squared
plus b x plus c equal to 0, that's the quadratic equation,
we're going to assume that a is not equal to 0, because if a is equal to 0,
then we know that we have b x plus c equal to 0, that's
what the equation will turn out to be, a is equal to 0, this becomes 0, so you have b x plus c equal to 0,
so you get x equal to -c divided by b, that's what you're going to get
for the solution.|So it is safe to assume that we are going to solve a quadratic equation
which only has . . . which has a not equal to 0, because we know for a equal to 0
the solution exists, which looks like that.|So let's . . . in assuming that a is not
equal to 0, we have a x squared plus b x plus c equal to 0,
and what I'm going to do is I'm going to divide by a
throughout, so I get this.
So this is . . . we obtained this by divide by a,
that's what I get, so if I divide by a both sides, so since I'm already assumed that a
is not equal to 0, so let me write this down again here, a not
equal to 0.|So I get . . . I get this particular formula.|Now what I'm going to do
is I'm going to add and subtract a quantity here, so I'm going to add
b squared by 4 a squared here, I'm going to subtract b squared
by 4 a squared here, and then plus b by a x . . .
sorry, plus c by a
equal to 0.|So what I'm doing is I am taking this quantity here,
adding it, and then I'm subtracting it immediately, and that gives me 0, so that reproduces the equation
which I had before there.|But what I can do is I can bundle this together.
I can bundle the first three terms of this term together, and they will turn
out to be x plus b divided by 2 a, whole squared, so that's
from your formula of a plus b, whole squared, that's what I get that from, and then I have this quantity
of minus b squared by 4 a squared plus c by a
equal to 0.|Having said that, what I'm
going to do is I'm going to take the last two terms of
the left-hand side to the right-hand side, so I'm going to take this to the right-hand side, I'll get b squared by
4 a squared plus . . . minus c by a . . . minus
c by a, that's what I get there.|And I'm going to
simplify this a little bit, I'm going to get 4 a squared here, and I'll do b squared
minus 4 a c here, so that's what it turns out to be.
And now what I'm going to do is I'm going to take the square root of both sides, I'm going to take the square root of both
sides here, and that's where I get the plus and the minus part of it there,
because when I take the square root of both sides, this can . . . this can be plus or minus.
So, let me write that down.|So I'll get x plus b divided by 2 a,
so that's taken by doing the square root of both sides, is equal to square
root of b squared minus 4 a c divided by
4 a squared, that's what I'm going to get from there.|Then what I can do is
I could write this as b squared minus 4 a c, divided by 2 a, because the
square root of 4 a squared is 2 a, so this should be plus and
minus, because both are possibilities when you take the square root of both sides,
so plus minus, I'm going to put plus minus there.|And then I get x is equal to
-b divided by 2 a, by taking this b divided by 2 a to the right-hand side,
plus minus square root of b squared minus 4 a c,
divided by 2 a, there.|And then I'm going to
write x is equal to, I'm going to take 2 a common in the denominator, so then I'll write
-b plus minus b squared minus 4 a c.
So that's the kind of form which you see as the solution for a quadratic
equation, which is the form a x squared plus b x plus c
equal to 0, is solution of
this.|So that's how we derive
the solution for a quadratic equation, the two roots of the
quadratic equation, a x squared plus b x plus c equal to 0.|And that's the
end of this segment.
.
.
