(male narrator)
In this video,
we will look
at finding a formula
that can help us
complete the square
on more involved problems.
This equation is
the general form of an equation
we would normally solve
by completing the square:
ax squared, plus bx,
plus c, equals 0;
where a, b, and c,
we will assume are numbers.
x is the variable
we are solving for.
To solve by completing
the square,
the first thing we do
is separate the variables
from the numbers.
Because c doesn't have
any x's on it,
we need to move it
to the other side
by subtracting c
from both.
This gives us ax squared,
plus bx, equals -c.
Our next step is we like
the x squared to be alone.
We need to divide by the a
to clear it out of the way.
As we do, we divide
each and every term by a,
giving us x squared, plus b,
over ax, equals -c, over a.
Next, we need to find the number
that completes the square.
We do this by taking half
of the middle term--
or the middle coefficient,
b over a--and square it.
This would be
b over 2a squared.
Squaring the numerator
gives us b squared,
and the denominator
gives us 4a squared.
This is what we have to add
on both sides of the equation:
b squared over 4a squared
on both sides.
When we've done this,
the left side of the equation
will be a perfect square.
We take the square root
of the first term, which is x;
the sign 
from the middle, plus;
and the square root
of the last term.
The square root
of b squared is b,
over 4a squared,
the square root is 2a.
On the other side, we need
to get a common denominator
in order
to add these together.
The common denominator
will be 4a squared,
so we need to multiply by 4a
on top and bottom.
That first fraction then
becomes -4ac, over 4a squared.
With the b squared on there,
all over the common denominator,
we have a b squared
and a -4ac,
all over the common denominator
4a squared.
We are now ready to get rid
of the exponent
by taking the square root
of both sides.
Square root and square
are inverses,
and we're left
with x plus b, over 2a,
equals the square root
of the fraction.
To take a square root
of the fraction,
we take the square root of the
numerator and denominator.
Don't forget we need
a plus or minus:
the square root of b squared;
minus 4ac;
which can't simplify,
because we can't take
the square root of a difference,
just a product;
over, the square root
of 4a squared is 2a.
To get the x alone,
we simply subtract b over 2a
from both sides.
When we do, because there's
already a common denominator,
we can write them
all over the single denominator.
We get the opposite of b,
plus or minus the square root
of b squared,
minus 4ac, all over 2a.
This result that we get for x
is going to be
very important to us
as we solve equations.
Anytime we have
the equation
ax squared, plus bx,
plus c, equals 0,
where we know what
the number a, b, and c are,
we can solve them
using this thing
called the quadratic...formula,
which says that x is equal
to the opposite of b,
plus or minus,
the square root of b squared,
minus 4ac,
all over 2a.
This formula
we will use a lot
and is very useful
to commit to memory.
A formula that we found
simply by completing the square
on ax squared, plus bx,
plus c, equals 0.
