OK.
Here's a method of solving
these quadratic equations that
will always work.
Now, I don't recommend that
this is your go-to method,
because it can be
a little tedious.
If you can solve it by
factoring or you can easily
isolate your x squared and
not have an x term in there,
I would do that, use the
square root property.
Otherwise, we'll use
the quadratic formula.
OK.
It says that I start
with ax squared
plus bx plus c equals 0.
Notice a couple things there.
First, it's all in
descending. ax squared,
then bx with a single exponent,
and then no x's at all.
The other thing is that
it has to be equal to 0.
I cannot use this formula
unless it's all equal to 0.
If that is the situation,
then these numbers in front,
the a, b, and c,
these coefficients
will help me find my answer.
We know that x is going
to be equal to-- this
is the opposite of b plus or
minus the square root of b
squared minus 4ac.
And notice it's all--
a big long line--
divided by 2a.
One thing I do recommend,
for sure, is this b
squared, that you do
that in your head.
You know that squaring means
multiplying a number by itself.
When you multiply a regular
real number by itself,
that's always going to
be a positive value.
The other thing is this
minus b out front here.
Remember, that means
the opposite of b,
so that's not always
going to be negative.
Let's take a look
at this example.
I think that will
illustrate it for us.
We're solving this one for x.
So 2x squared minus
4x plus 7 equals 0.
First, identify a, 2.
b, it's the number in
front of x, including
that sign, so negative 4.
And then c is going to
be this constant here,
which is a positive 7.
So if I put it in
my formula, x is
going to be equal to
the opposite of b.
So notice it's a positive 4,
plus or minus square root,
and then I take b negative
4 multiply by negative 4.
Get positive 16 minus
4 times a times c.
All divided by 2 times a.
OK.
Now it's just some arithmetic
and multiplying stuff out.
So you get 16 minus
56 underneath,
which gives me a negative 40.
OK.
Here's a little
bit of a problem.
We noticed in the
previous section
that when I have a square
root of a negative,
that really means I have
an imaginary number.
So let's break down the
radical on the side here.
The square root of negative
40 could be broken down
as the square root of negative
1 times the square root of 4
times the square root of 10.
We can simplify the
square root of negative 1
as our imaginary number i.
The square root of
4 we know to be 2,
and then our square root 10.
This is still multiplication.
And normally, we put
numbers in front of letters.
So I'm going to write
that as 2i root 10.
And that's what shows
up here in my next line.
So I've got 4 plus or minus
2i root 10 all divided by 4.
Now, before I put that
in as my final answer,
I'm going to look at this
and say, wait a minute.
All these numbers out here--
I don't worry about
what's under the radical--
all these numbers out here,
they're all divisible by 2.
If I can divide all three
of these by the same value,
I should do that.
That's going to be
my simplified answer.
So that's where
this final answer
of 2 plus or minus i root 10,
all divided by 2 comes from.
All right.
Let's try another one.
What if I have 3x squared
plus 6 equals 10x?
Well, the first red flag is,
hey, it's not equal to 0.
So that means we're going
to have to subtract this 10x
and move it to the other side.
But I have to remember
to be careful here,
because when I move
it to the other side,
I have to keep my
powers in order.
So I'm going to put 3x squared
minus the 10x then the plus 6
equals 0.
So then I can identify
a is 3, b is right
your front of the
x, the negative 10,
and c is always the
plain constant term of 6.
Again, we put it into
our formula, the opposite
of b plus or minus
my b squared, which
I did in my head, 10 times 10,
negative 10 times negative 10,
minus 4 times 3 times
6 all over 2 times 3.
Do your arithmetic,
and this time,
we get a positive number
underneath our radical,
the square root of 28.
OK.
I didn't show you the
scratch work here,
but it's pretty easy.
28 is the same as 4 times 7.
And I know the square
root of 4 is 2.
So that's where
that 2 came from.
Once again, all these
numbers outside the radical
are divisible by
2, so I reduce it.
OK.
Now I want to show you
this really cool trick.
I have just a minute here.
If I take the one
that's equal to 0,
and I put that
into my calculator,
I can check my answer.
So 3x squared minus 10x plus--
oops, back up-- plus 6.
Even though it's a
messy-looking answer,
I'm going to check this.
Some graphing it here
in the standard window.
And remember that if
I set it equal to 0,
then what I'm checking
for are these intercepts.
Well, I don't even have
to go through and make
the calculator find the 0.
I'm just going to
put in a value.
So I'm going to work
with a positive 1.
I want the numerator
here, 5, plus--
and I'm going to use
my square root key--
square root of 7.
Close up the radical.
Close up our numerator.
I want all of that--
that's why there's
parentheses-- divided by 3.
I'm going to see
what it tells me.
Yes!
y is equal to zero.
A little hint, if
the positive 1 works,
the negative 1 will also.
