In our last few
examples, we have
been solving equations
of the form ax
squared plus bx plus c equals 0.
It has taken us quite a
while to do each example,
and it's taken a lot of steps.
But notice that every time we
do it, the steps that we take
are just about the same.
So let's try to do it for
general numbers a, b, and c
so that we don't have to
do these steps every time.
The first thing that we did was
write this in the form a times
x minus h squared plus k.
Please click the
link now to see how
we can write it in this form.
It was very messy,
but we found that we
can write ax squared
plus bx plus c
as a times x minus
negative b over 2a
squared plus 4ac minus
b squared all over 4a.
Now, to solve for x in a times
x minus negative b over 2a
squared plus 4ac minus b
squared over 4a equals 0,
we have to undo four operations.
They're just messier
than they were before.
But first thing we do to x is
we subtract negative b over 2a.
The next thing we
do is square it.
After that, we multiply by a.
And finally, we
add the number 4ac
minus b squared all over 4a.
As we have done when we knew
the numbers a, b, and c,
we want to undo these
in the opposite order.
So to undo adding 4ac
minus b squared over 4a,
we subtract that number.
And to undo multiplying
by a, we divide by a.
Undo squaring, we
take the square root.
And to undo subtracting
negative b over 2a,
we add negative b over 2a.
So let's start by
subtracting 4ac minus b
squared over 4a from both sides.
This will give us a times
x minus negative b over 2a
squared plus 4ac
minus b squared over
4a minus 4ac minus
b squared over 4a
equals 0 minus 4ac
minus b squared over 4a.
We can simplify the right
a bit by distributing
that negative sign, making
negative b squared positive
b squared and 4ac negative 4ac.
And on the left, we have that
these two numbers cancel out,
because one is positive
and one is negative
no matter how messy they are.
So on the left, we have a times
x minus negative b over 2a
all squared.
The next thing we have to do
is divide by a on both sides.
This gives us a over x
minus negative b over 2a,
all squared, divided by
a equals 1 over a-- this
is another way we can
divide by a-- times
b squared minus
4ac, all over 4a.
On the right, a times 4a in
the denominator is 4a squared.
While in the numerator,
we're just multiplying by 1.
So we have b squared
minus 4ac still.
And on the left, a
divided by a is 1.
So we have x minus negative
b over 2a, all squared.
Now we need to take
square root of both sides.
Doing that gives
the square root of x
minus negative b over
2a squared equals
the square root of b squared
minus 4ac over 4a squared.
But if we take the square
root of a fraction,
we can take the
square root of the top
and take the square root
of the bottom separately.
In the top, we have the square
root of b squared minus 4ac.
While in the bottom,
we have the square root
of 4 times a squared.
But we can reduce that, because
the square root of 4 is 2,
and the square root
of a squared is a.
So we have the square root of b
squared minus 4ac all over 2a.
But on the left, we
have the square root
of x minus negative b
over 2a, all squared,
which is the absolute value
of x minus negative b over 2a.
So we have that
the absolute value
of x minus negative b over 2a
equals the square root of b
squared minus 4ac, all over 2a.
But addressing the
absolute value,
we get that x minus
negative b over 2a,
the number inside
the absolute value,
has to either be plus or
minus the square root of b
squared minus 4ac all over 2a.
And finally, to get just x, we
need to add negative b over 2a
to both sides.
This gives x equals
negative b over 2a
plus or minus the square root
of b squared minus 4ac over 2a.
And since these have a
common denominator of 2a,
we can combine the
tops to get that x
is negative b plus or
minus the square root of b
squared minus 4ac, all over 2a.
This means that no matter
what a, b, and c are,
we can solve for x without
doing all the steps that we've
been doing, but simply
using this formula.
So in trying to find the
solutions to ax squared
plus bx plus c
equals 0, we found
that the solutions
are given by x
equals negative b plus or
minus the square root of b
squared minus 4ac, all over 2a.
This formula for solving
quadratic equations with 0
on one side has a famous name.
This is called the
quadratic formula.
And using this, we can
now compute solutions
to every quadratic
equation very quickly,
much quicker than
we could before.
So while it was very hard
to come up with this,
it'll help us in the long run.
