Today we're gonna go over the second great
thing every student knows in mathematics:
The Quadratic Formula
You know the one:
minus b plus or minus square root ... wait
wait wait ...
If you're like me, you hate this thing
If you're not like me, you probably still
hate it, but for different reasons.
Let's have a look at an alternative way to
do the exact same thing that is a million
times less ... THIS.
We're going to be shortening the idea from
3blue1brown's recent video on this topic,
which itself is based on a video by Po-Shen
Loh. Both are linked in the description.
To start off, we want to talk about quadratics
that look like
x^2 + b'x + c' = 0
We could instead discuss quadratics like
ax^2 + bx + c = 0
but those aren't as easy. And besides, if
all we want is an equation to equal zero,
wouldn't it be nice to work with a simpler
one?
We can justify leaving off that piece in two
parts
First, we can divide the whole equation by
a to get a thing that looks like this:
x^2 + (b/a)x + (c/a) = 0
Second, we can show on a graph that, as long
as we don't multiply or divide by
zero, the places where the expression is zero
don't change when multiplying.
Okay, great, so we start with
ax^2 + bx + c = 0,
divide by a and re-write
x^2 + b'x + c' = 0
Now we can get started. We turn back to the
graph for a moment.
We're going to do some steps here.
First, those zeroes are gonna be called R
and S
Second, we can write an equation using R and
S that represents our situation
(x-R) (x-S) = 0
Third, we do that weird multiplication thing
to get
x^2 - (R+S)x + RS = 0
This tells us two important facts. Namely,
b' = -(R+S)
c' = RS
in our simplified equation.
Fourth, we identify the middle of R and S
... call it m
We know the zeroes are the same distance from
m ... call it d
But that means the product RS can be written
as
(m-d)(m+d)
We do that weird multiplication thing again
to see that
RS = m^2 - d^2
but wait ... c' = RS so c' = m^2 - d^2
Or in other words, d = square root (m^2 - c')
AND this gets us to the point we needed to
be
Why? Because R and S are our zeroes
but we can write those together as
x = m plus or minus d
HEY! We know how to find d, so let's put that
in there
Now we know c', and the only thing we're missing
is m...
Wait! m was the midpoint. We can write that
as the average of the two points...
so m = (R+S)/2
We've seen something like that already...it
was in b'
Great, so we can write the whole solution
like this:
If x^2 + b'x + c' = 0
Then x = m plus or minus square root m^2 - c'
where m = -b'/2
What's easier to remember? This or THAT?
Before we leave off, let's see this in action.
Suppose we want to solve 2x^2 + x - 6 = 0
We divide off the 2 to make the equation a
bit nicer.
Now we have x^2 + x/2 - 3 = 0
Under our labels from before, b' = 1/2 and
c' = -3
So m = -b'/2 OR just -1/4
And our answer is
x = m plus or minus square root m^2 - c'
OR
-1/4 plus or minus square root 1/16 + 3
Simplifying, this is
-1/4 plus or minus square root 49/16 or -1/4
plus or minus 7/4
Then our answers are -2 and 3/2
Now tell me ... which way would YOU prefer?
