- So let's consider five X squared
minus 11 X plus eight equals zero.
We're going to use the
quadratic formula to solve this.
We see that solutions of A X squared
plus B X plus C equals zero,
are given by X equals negative B
plus or minus the square root
of B squared minus four AC,
all divided by two A.
So you can google the quadratic formula.
You can hear lots of different
quadratic formula songs,
to the tune of pop goes the weasel.
Okay, so is our equation
in this form?
Okay, well, we have some
term with X squared first,
and then we have a term with an x,
this term with an X , we
need the constant, so yes.
So let's figure out what
each letter represents.
AX squared plus BX plus C equals zero.
So A is the coefficient of X squared,
so A is five.
B is the coefficient of X,
so B is negative 11,
and C is our constant.
So C is eight.
Now we're going to plug each
piece into this equation.
So X equals negative B plus
or minus the square root,
of B squared minus four AC,
all divided by two A.
So let's just highlight each piece.
So,
A is five,
so we're going to put five in that place.
B is negative 11,
So put a negative 11,
just for b, in right here,
and in the C, is eight,
so we gonna plug this in.
So X equals the negative of negative 11,
excuse me,
plus or minus the square
root of negative 11 squared.
Minus four times A,
so four times five times C,
and C is eight,
all divided by two times
A or two times five.
So let's go through this
and make sure we got it.
So negative of negative 11,
and then we have the negative 11 squared,
because we have B squared over there.
A is 5, so here's my A,
and then my C is eight.
So take just a moment
and make sure negative B
plus or minus the square root of B squared
minus four times A times C,
all of it divided by two A.
So let's just start cleaning it up.
Well, the negative of negative 11,
is the same as just positive 11,
because you negate the negative,
and you get the opposite
sign, you get a plus.
Gives you 11, plus or
minus the square root
well, negative 11 squared,
be very, very, very careful here.
Two things are being square here.
The 11 and the negative sign,
so this turns into a positive 121.
Now Be very careful,
negative 11 squared is not the same thing,
as negative 11, squared.
When you're writing this,
B squared under your root,
make sure you put the
whole thing in parentheses.
This,
is negative,
of this.
So negative 121.
This is positive 121.
They're not the same thing.
Make sure you're very, very,
very, very, very, very careful,
and you use these parentheses right here.
Okay.
So then we have negative
four times five times eight.
So that is 160.
Be very careful here.
Check your sign.
You have A negative times two positives,
you're ending up with a negative 160,
and all of it is divided by two times five
or two times five is 10.
Okay.
So we have X equals 11 plus
or minus this square root.
So we have 121 minus 160,
we're going to get a negative number,
we get negative 39.
All of that's over 10.
Okay, we know already that
that's not a real number,
you have an imaginary number, recall,
the square root of negative
one is defined to be i.
So here we have the square
root of negative 39.
We have a rule from algebra that says
the square root of A times B,
is equal to the square root of
A times the square root of B.
So this can be rewritten
as a square root of
negative one times 39,
but that is the same as the
square root of negative one
times the square root of 39,
and then we see that this
square root of negative one,
is equal to i right there.
So we have X equals 11 plus or minus
square root of 39 times i over 10.
So there are two non-real
solutions.
The complex numbers are X equals 11
plus the square root of 39.
I divide each piece by 10
if you wanted it in A plus Bi form,
or X equals 11 minus,
so we're going to 11 over 10
minus square root of 39 over 10 times i.
So these are both complex numbers.
Let's do another example.
Let's fill it in right here.
Let's see.
Let's also consider,
let's do two X squared minus three X
minus eight equals zero,
and let's solve
with
the quadratic formula.
So we're going to follow
this pattern over here a bit.
We're going to start fresh
with different variables.
So,
different constants rather,
and coefficients.
So A,
is the coefficient of X squared.
I used two,
and we see that B is the coefficient of x.
In this case B is negative three,
and we see that C is the constant term,
so C is negative eight.
Then we fill in X equals negative B
plus or minus the square root
of B squared minus four AC all over two A,
and if you are not
familiar with this formula,
I recommend that you write it
down every single time you use it.
The more you write it down,
the more problems you work,
the more comfortable you're going to be,
and you'll have this formula
memorized in no time.
So X equals the negative
of negative three,
plus or minus the square root
of negative three squared.
Notice here again,
just like over this example,
I'm putting parentheses
because I want to square
the three and the negative sign, okay?
Minus four times A,
times C,
all of that is over two
times A or two times two.
So let's just color code this,
and make sure you can
see what we're doing.
There's your C,
your B was negative three,
so negative B,
and then we have B squared.
Okay,
and then we have A,
is
your two.
Okay, got it?
So then we have the
negative of negative three.
So X equals positive three,
plus or minus.
Well in here we have
negative three squared,
so that's three squared,
and also the negative squares,
we ended with a positive nine.
Now we have a four times
A two times an eight,
if you just look at the numbers.
So four times two is eight,
eight times eight is 64.
Now let's check the signs of
a negative times a positive.
So overall, the product
here is a negative,
and then multiplied by another negative,
we end up with a plus.
So we end up with
X equals three,
plus or minus the square
root of 73 over four.
So this one actually has real solutions.
These are real solutions.
We cannot break square
root of 73 down further,
it doesn't have any
perfect square factors.
So we're going to say
this is X equals three
plus square root of 73,
all of it divided by four,
or X can be three minus
the square root of 73,
all of it divided by four.,
and we see these are
what we call irrational
solutions.
They're real numbers,
but they're not integers
and they're not rational numbers.
Rational means fraction, remember.
