>> I want to start this second by
deriving the quadratic formula,
because so often students know
what the quadratic formula is,
but they don't know where it comes from.
So, let's go through and derive this formula.
The formula is used when we're
solving quadratic equations
in the form ax squared plus bx plus c equals 0.
And the quadratic formula is
essentially a shortcut for going
through the process of completing the square.
Remember, when we were completing the
square, when there was a leading coefficient
on that x squared, the first step
was to get rid of it by division.
And that's what we're going to do here.
We're going to divide both sides by a.
When we do that, we have x squared plus b
over a times x plus c over a equals
0, because 0 divided by a is 0.
We're assuming a is not 0.
Now, the next step before completing the
square, we wanted to move that constant term
to the other side, just to
get it out of the way.
So, here, we're going to subtract c over a on
both sides, which gives us x squared plus b
over a times x equals the opposite of c over
a. And now we're ready to complete the square.
The coefficient on the x is given
by b over a. If we divide that by 2,
it's the same as multiplying by 1/2.
So, b over a times 1/2 is going to be b over 2a.
And then if we square that, well,
the numerator gives us b squared.
And in the denominator, when we take
2a and square that, we get 4a squared,
because 2 squared is 4, and
a squared is a squared.
But we know that if we add something
to the left hand side of an equation,
we also have to add it to the right hand side.
So, we need to add b squared
over 4a squared to both sides.
Now that we have completed the square,
we're able to factor the left hand side.
When we do that, we end up with x
plus b over 2a quantity squared.
And then on the right hand side,
we need to add those fractions
by coming up with a common denominator.
So, we've got negative c over a
plus b squared over 4a squared.
The common denominator here
is going to be 4a squared.
Now, the second fraction already
had the common denominator.
So, because the denominator didn't change,
the numerator is also not going to change.
We're going to have that b squared there.
But in the first one, to get from a to
4a squared, we have to multiply by 4a.
So, we're also going to multiply
the numerator by 4a.
That's going to give us 4ac, still
with that negative sign out front.
And if we want to add these, then we
need to add the terms in the numerator.
I'm actually going to switch the order.
I'm going to put the b squared first and then
the minus 4ac after that all over 4a squared.
And so when we add those fractions together,
we get b squared minus 4ac over 4a squared.
Now, if you're looking up at the quadratic
formula, there's a b squared minus 4ac up there,
and we've got a b squared minus 4ac down here.
So, we're on our way to getting there.
So, at this point, we have a quantity
squared equals what is effectively a number.
All of these variables represent
constants in the original problem,
if we're actually solving a quadratic equation.
So, all of this ends up being a number.
And so we can use the square
root property at this point.
When we take the square root
of the left hand side,
that effectively undoes the squaring
and gives us x plus b over 2a.
And on the right hand side, we
have plus or minus the square root
of b squared minus 4ac over 4a squared.
Now, the numerator is not a perfect
square, but the denominator is.
And because this is in a square
root, we can actually evaluate that
and take it out of the square root.
This gives us x plus b over
2a equals plus or minus.
In the numerator, we're still going to have
the square root of b squared minus 4ac.
But in the denominator, we can evaluate the
square root of 4a squared to give us 2a.
I'm going to rewrite what we have over
here, just to have some more room.
X plus b over 2a equals plus or minus the
square root of b squared minus 4ac over 2a.
And the last step in getting x
alone is to get rid of this term b
over 2a by subtracting it on both sides.
When we do that, we're left with x on the left,
and on the right we've got the opposite of b
over 2a, plus or minus the square
root of b squared minus 4ac over 2a.
And because we have two fractions with
the same denominator on the right,
we can combine the numerators into one large
fraction, which gives us the opposite of b plus
or minus the square root of b
squared minus 4ac, all over 2a.
And this is the quadratic formula.
And we got this by going through the
process of completing the square.
But now that we've done this, we will not
have to complete the square any more times.
Instead, to solve a quadratic equation,
we look at the coefficients a, b and c,
we substitute them into this
formula, we go through the process
of evaluating whatever this is, which we
will get practice with in many problems
in this section, and then that
will give us the answer for x.
