>> This is Part 1 of
solving equations using the
quadratic formula.
And we solve this equation
using the quadratic formula.
Also we do it by factoring
and we check the solutions.
This is Part 1 of using
the quadratic formula
to solve equations.
You should have already
watched videos
on using the square
root property
and completing the square and
deriving the quadratic formula.
So here's the quadratic
formula equals
if AX squared plus
BX plus C equals 0,
then X equals negative B
plus or minus the square root
of B squared minus
4AC all over 2A.
So if we could put
something in this form,
AX squared plus BX
plus C equals 0,
then we can extract
the values for A, B,
and C and plug it
into that formula.
So here's an example.
5X squared equals 14X plus 3.
The first step is to rewrite
this so it's set equal to 0.
So 5X squared minus 14X minus 3
equals 0, and then identify A,
B, and C. A is the
coefficient of X squared,
B is the coefficient of
X, and C is the constant.
Okay. So here was our problem
after we set the problem equal
to 0 to put it in the form.
Again, we would say what
A, B, and C are equal to.
And in the formula,
we're just going
to go plug in those numbers.
So I'm going to plug
in the As, Bs, and Cs.
So I've got negative B,
so I could write negative.
And, then, for B, you can
plug in negative 14 plus
or minus the square root.
And then we can put
negative 14 over B here,
negative 14 squared minus
4 times A, which is 5;
and then C, which is negative 3.
So that would be
the whole numerator.
And the denominator is
2 times A, which is 5.
So you could just simply
completely just plug
in all those As, Bs
and Cs directly in.
But you do have -- as you see,
it's kind of a mess
of numbers here.
And that's why I
like breaking it
down a little bit differently.
And now I'm going to take
the opposite of negative 14,
which is positive
14 plus or minus.
And now, underneath the square
root, we have to simplify.
So first we have
to do the squaring,
so negative 14 squared, again,
is negative 14 times
negative 14.
That's 196.
And then I've got minus 4
times 5 times negative 3.
So we have a negative and a
negative, and that's going
to be a positive; and
that will give me 60.
And, then, the bottom,
I have it all over 10.
And you still have to simplify
inside the square root,
which would be 256 all over 10.
Then we want to write
square root
of 256 is 16 just all over 10.
So I finally get -- that's
what X equals, right?
So then I could do X
equals 14 plus 16 over 10.
And we could simplify that going
down the page, if you want.
30 over 10 equals 3 or X
equals 14 minus 16 over 10,
which is negative two-tenths
or negative one-fifth.
Take a look again at how
I began this problem.
First I set the equation equal
to 0 to get it in standard form.
I stated the values of A,
B, and C. And then I plugged
in the values of A, B, and C
into this quadratic formula.
And it's quite a fraction.
And sometimes people
make mistakes
when they see all
of this all at once.
So an alternative way to do it,
the way I like to do it is I
like to figure out what B
squared minus 4AC is first.
So B squared minus 4AC, I just
like to figure out what's going
to go under the square root.
And I look up here and
I see B is negative 14.
So I do that in my head.
What's negative 14
times negative 14.
And, of course, you
could use a calculator.
And you're going to get 196.
And then it says minus 4AC.
I like to do AC in
my head as well.
5 times negative
3 is negative 15.
So do you see how this would
be B squared minus 4AC.
So I'm doing the B squared,
negative 14 times negative 14.
And I'm doing the AC,
5 times negative 3.
And now we simplify that.
That's 196 plus 60,
which is 256.
All right.
So we sort of worked
that out ahead of time.
I think you'll see this makes
it a little less cumbersome
to plug in the formula.
So now the formula says it's
the opposite of B. All right.
So if B is -- in fact, I'm going
to write the formula again.
Negative B plus or
minus the square root
of B squared minus
4AC all over 2A.
So now I'm going to replace
negative B. I'm going to put
in the opposite of
whatever B is.
So if B is negative
14, then the opposite
of B is positive 14, okay?
Plus or minus.
Now, it's just the
square root of 256.
Now, if it's a perfect square,
let's write down
what that would be.
What would the square
root of 256 be?
Make sure you realize that's
what would be underneath the
square root.
And that's 16, all over 2A.
A is 5, so 2 times 5 is 10.
So what I'm doing is
I'm showing you how --
instead of showing
every single step here,
it's easier to do a few
of these in your head.
Otherwise, it gets
kind of crazy looking.
But I'll do it the
long way in a minute.
All right.
So this gives us
two possibilities.
X can either be 14 plus 16 over
10, which is 30 over 10 or 3;
or X can be 14 minus 16 over
10, which is negative 2 over 10
or negative one-fifth.
So the solutions, if I
didn't make any mistakes,
are 3 and negative one-fifth.
First of all, if you use
the quadratic formula
or completing the square
and you get answers
that don't have any
square roots,
that means you could have
also done the same problem
by factoring, find the
solutions by factoring.
So for now, though, let's write
down what these solutions are.
The solutions are 3
and negative one-fifth.
Okay. This was our
original problem.
These were our solutions.
Let's see if we would have
gotten the same answer
by factoring.
So I'd have 5X squared
minus 14X minus 3 equals 0,
and then we try factoring.
So I do it by trial factors,
but you can use any
method you want to factor.
Let's see.
It's going to be 3
and 1 minus and plus.
That's how to factor 5X
squared minus 14X minus 3.
And then we would set each
factor equal to 0 and solve.
So notice we still get X
equals negative one-fifth
and X equals 3.
And for this particular problem,
we could have found the
solutions this way as well.
Now, the other way to check
your answers, of course,
is to plug in these
values one at a time
into the original equation.
So let's do that.
We're going to take
the original problem,
5X squared equals 14X plus 3.
And first we'll plug
in 3; then we'll plug
in negative one-fifth.
All right.
First let's plug in 3.
Now, you of course, can
put the video on pause
and try this on your own.
So we're going to put in 3 for
X, so that becomes 3 squared.
And remember, order
of operation,
you have to do 5 times 9.
You have to do the
exponents first.
And 5 times 9 is 45.
On the other side, we
have 14 times 3 plus 3.
So we have 42 plus
3, which is 45.
So 3 also checks.
And now let's go ahead and
check the negative one-fifth.
This is one will be a
little bit more involved.
So we're going to plug in
negative one-fifth for X.
So we have 5 times -- all right.
When you do negative one-fifth
times negative one-fifth,
that's one 25th.
And then we can cancel.
So this side reduces
to one-fifth.
And on the other side,
we have 14 times --
times negative one-fifth plus 3.
So we have negative 14 fifths.
And I have to get a
common denominator for 3,
so I need to write
that as fifths.
And that would we
15 fifths, right?
You multiply numerator
and denominator by 5
to get 15 fifths, and negative
15 fifths plus 15 fifths
is one-fifth.
So I got the same
number on both side.
So that doesn't mean
one-fifth is the answer.
It means that we checked,
and negative one-fifth
was the correct answer.
All right.
So we did it by using
the quadratic formula,
we did it by factoring, and
we checked both solutions.
