- [Instructor] Solving
by the quadratic formula.
So use the quadratic formula to solve
six x squared plus seven
x minus three equals zero.
And we see over here, this
is the quadratic formula.
So a lotta students
leave off a few things.
They may get this part right,
but they leave off that
this is exactly solutions
for a x squared plus
bx plus c equals zero.
So this is an equation in quadratic form.
It has to be in this form.
All the terms have to be
on the left-hand side.
You can't have a x squared plus bx equals
some other number over here.
Everything has to be on the left.
And if this is your equation,
the solutions are x equals this.
So a lotta students leave off the x
and they forget you're
talking about x-values
when you finish.
So solutions of quadratic
equations are x equals
negative b plus or minus the square root
of b squared minus four
ac, all over two a.
Now if you wanna Google this,
you can Google the quadratic formula song
and then you'll actually,
it's a wealth of options.
There's many, many versions.
You can hear people singing
the quadratic formula
to the tune, excuse me,
of "Pop Goes the Weasel."
There's also an Adele
song that's been modified,
lots and lots and lots of things.
However you wanna remember it,
this is very, very, very important.
You really need to know how to
solve a quadratic equations.
So if we're gonna use
the quadratic formula,
it would be helpful
if we know what these letters were, right?
So label a few things.
a x squared plus bx plus c equals zero.
I'm just rewriting the
form above our equation.
So we're looking for a.
a is the coefficient of x squared.
So look at our equation.
We see x squared.
Coefficient, remember, is
just the number multiplied
in front of the variable.
So if we have x squared,
the coefficient is six.
Okay, and then we see that b
is the coefficient of x.
So we have seven x.
So b must be seven.
And we see that c is the constant.
So just remember, constant
means there's no variable there.
It's really like x to the zero.
So our constant term is negative three.
And then, we're going to plug
in all of the pieces, okay?
So x equals negative b plus
or minus the square root
of b squared minus four ac,
all of it divided by two a.
So a couple things I wanna point out.
Just be very careful.
Please memorize this formula.
Please commit it to memory.
Biggest mistakes I see is students forget
the sign in the middle.
A lot of 'em only divide
one piece by two a.
It's all of it divided by two a.
So let's fill in each piece.
The next biggest mistake
I see is people fill in
things other than a, b, and c.
The only place you see an
x is on the left-hand side.
There are no variables over here
on the right-hand side, okay?
Only plugging in variables
over here with x, okay?
So we see a is six.
We're gonna put it right there.
b is seven, so you're gonna
put a seven right there.
c is negative three.
So let's fill it in.
x equals negative b, so negative seven,
plus or minus the square root,
okay, then we have b squared.
Well, b is seven, so this is seven squared
minus four times six times negative three.
And all of that is divided by two times a,
so two times six.
Just taking a moment to color code this
so you can see what we've done.
Again, please be very, very, very careful.
A lotta students wanna put x's in here.
No x's over here, just the
numbers, the coefficients
or the constant go on this side.
Then we start cleaning it up.
x equals negative seven
plus or minus the square root of, well,
seven squared is 49,
and then we have four,
negative four times six
times negative three, okay?
And six times four is
24 times three is 72.
But notice, wanna point out something,
be very, very, very careful.
You have a minus sign here
and a minus sign here.
A negative times a negative
becomes a positive.
Okay, and then we divide by 12.
So then, we're going to
clean up under our root.
So x equals negative seven plus or minus
the square root of, we have
49 plus 72, which is 121,
and divide the whole thing by 12.
Well, the square root of 121 is 11,
so we have negative seven plus or minus 11
over 12.
We can write this as two answers.
So x equals negative seven plus 11 over 12
or x equals negative
seven minus 11 over 12.
Well, negative seven plus 11
is the same as 11 minus seven, right?
So we get four over 12 and notice the sign
that's actually in here is a positive.
So we can reduce four over 12,
divide both pieces by four,
the top by four, you get one,
divide the bottom by
four and you get a three.
Then over here we have
negative seven minus 11.
If you have two numbers
that you're combining
with the same sign,
you're going to add them
and then use the sign they both have.
So seven plus 11 is 18,
so you get negative 18 over 12.
And then, divide each of these
by, what, six, I think so.
18 divided by six, you
get a negative three,
12 divided by six, you get a two, okay.
Perfect.
So there are two answers, x equals 1/3
and x equals negative 3/2.
Let's try another one.
Use the quadratic formula to solve
three x squared plus four x equals one.
So let's label each piece.
a is the coefficient of
x squared, so it's three.
Is the coefficient of x, which is four.
What's c?
You may be tempted to say that c is one.
c is not one.
Why is c not one?
We haven't used that number yet, right?
Okay, c is not one
because we do not have our
form written correctly.
So it says solutions of a
x squared plus bx plus c
equals zero are given by
this quadratic formula.
We don't have that, right?
We have a x squared plus bx,
there's no plus c, right?
So we probably need to rewrite this.
So we need three x squared plus four x.
The problem is, this
constant should be over here,
right, 'cause we want plus c equals zero.
So we're going to subtract
one from both sides.
So three x squared plus four
x minus one equals zero.
Now we can see, (laughing)
we can see what c is.
c is negative one.
Okay, so then, solutions are
x equals negative b plus
or minus the square root
of b squared minus four
ac, all over two a.
So negative b is negative
four plus or minus
the square root, b squared is four squared
minus four times a times c,
and the whole thing is
divided by two times a, okay?
So we get negative four plus or minus
the square root of,
well, four squared is 16.
And here, be very careful.
You have negative four times a three
times a negative one.
So three times four is 12, right?
If you have a negative times
a negative, you get a plus.
Plus 12 and the denominator is six.
So far, so good?
Okay, so then we have,
oh let me just go and do it right here.
We have negative four plus
or minus the square root of,
we have 16 plus 12, so
that's what, 28, over six.
Well, 28's not a perfect square,
so we're gonna take just a
minute and we're gonna look
at the prime factorization
of 28 to pull out
any perfect square factors.
Okay, so we're just gonna do it over here.
28, we factor as two times 14.
And then 14, we factor
further as two times seven.
And we see we have the prime factors two
and two and a seven.
So the square root of 28 is the same thing
as the square root of two
times two times seven.
Well, when you have two of something,
two factors under a root,
especially if it's a square root,
there's an understood little
two right here, right,
you need two of them,
a pair of them to come out as a single.
So like the square root of four is a two.
So that's what this is, right?
So this is really like let
me write it a little bit
one more step, this is
the same thing as writing
the square root of two times two
times the square root of seven.
Because we have a rule
from algebra that says
the square root of ab is the same
as the square root of a
times the square root of b.
This is square root of four,
so this is two root seven.
Two square root of seven over six.
We would like to clean
this up to simplify this.
So we have negative four
plus or minus two root seven over six.
We notice every single one of these,
the four, the two, and
the six, are all even.
So we can factor out a two.
If we factor out a two,
we get negative two,
'cause two times negative
two gives you negative four,
plus or minus, well, if
we have two root seven
and we factor out a two,
we're just gonna get
plus or minus root seven,
and the bottom we still have our six.
So we can take this two and this six
and we can reduce them.
We can divide both of them by two.
So two divided by two is a one.
Six divided by two, we
have a three on the bottom.
Y'all check, make sure that's right.
So negative two plus or minus
the square root of seven over three.
And remember that's x equals.
There are two answers.
x equals negative two plus
the square root of seven,
the whole thing divided by three,
or x equals negative two
minus the square root
of seven over three.
Those are the possible solutions.
