CA Algebra I: Quadratic Equation

53-57, Quadratic Equation

We're on problem 53.

It says Toni is solving this
equation by completing the

square. ax squared plus bx plus
c is equal to 0, where a

is greater than 0.

So this is just a traditional
quadratic right here.

And let's see what they did.

First, he subtracted c from
both sides and he got ax

squared plus bx is
equal to minus c.

OK, that's fair enough.

And then let's see.

He divided both sides by a.

Right, that's fair enough.

He got minus c/a.

Which step should be Step
3 in the solution?

So he's completing the square.

So essentially, he wants this
to become a perfect square.

So let's see how
we can do that.

So we have x squared plus b/a
x-- and I'm going to leave a

little space here-- is
equal to minus c/a.

So for this to be a perfect
square we have to add

something here, we have
to add a number.

And we learned from several
videos in the past and we kind

of pseudo-proved it.

And actually, I have several
videos I do solely on

completing the square.

You essentially have to add
whatever number this is, add

half of it squared.

And if that doesn't make sense
to you, watch the Khan Academy

video on completing
the square.

But what is half of b/a?

Well it's b over 2a.

So 1/2 times b/a is equal
to b over 2a.

And then, we want to
add this squared.

So let's add that to both
sides of this equation.

So we're left with x
squared plus b/a x.

And we want to add
this squared.

Plus b over 2a squared is
equal to minus c/a.

Anything you add to one side of
the equation, you have to

add to the other.

So we have to add that
to both sides.

Plus b over 2a squared.

And let's see if we've
solved the problem so

far, what they want.

X, b over 2-- right.

This is exactly what we did. x
squared plus b/a plus b over

2a squared, and they add it to
both sides of the equation.

So D is the right answer.

Now if you find that a little
confusing or if it wasn't

intuitive for you, I
don't want you to

memorize the steps.

Watch the Khan Academy video
on completing the square.

Next problem, 56.

No, 54.

All right, this is another one
that should be cut and pasted.

All right, four steps to derive
the quadratic formula

are shown below.

I said in previous videos that
you can derive the quadratic

formula by completing
the square.

And we actually do that
in another video.

I don't want to give too much
of a plug for other videos,

but let's see what
they want to do.

What is the correct order
of these steps?

So the first thing you want to
start off with is just a

quadratic equation.

And this one is the
first step.

This is where we started off
with in the last problem.

Then what you want to do is
add 1/2 of this squared to

both sides.

So b over 2a squared you want
to add to both sides, and

that's what they did here.

So our order is I.

And then you want to do IV.

That's what we did in
the last problem.

We did IV.

And then from here, you know
that this expression right

here is going to be equal to
x plus b over 2a squared.

And once again, watch soon.
the completing the squared

video if that didn't
make sense.

But the whole reason why you
added this here is so that you

know that, OK, what two numbers,
when I multiply them

equal b over 2a squared, and
when I add them equal b/a?

Well that's obviously,
b over 2a.

If you add it twice you're
going to get b over a.

If you square it, you're going
to get this whole expression.

So you say, oh, this is just x
plus b over 2a squared and you

get that there.

And then, is equal to--
and then they just

simplify this fraction.

They found a common denominator
and all the rest.

And so the next step
is Step II.

And then all you have
left is Step III.

And you've pretty much derived
the quadratic equation.

So I, IV, II, III.

That's choice A.

Problem 55.

Which of the solutions--
OK, I'll put all

of the choices down.

So which is one of the solutions
to the equation?

So immediately when you see all
of the choices, they have

these square roots
and all that.

This isn't something that
you would factor.

You would use a quadratic
equation here.

So let's do that.

So the quadratic equation is, so
if this is Ax squared plus

Bx plus C is equal to 0.

The quadratic equation
is minus b.

Well they do it lowercase.

Plus or minus the square root of
b squared minus 4ac, all of

that over 2a.

And this is just derived from
completing the square with

this, but we do that
in another video.

And so let's substitute it in.

What is b?

b is minus 1, right?

So minus minus 1, that's
a positive 1.

Plus or minus the square
root of b squared.

Minus 1 squared is 1.

Minus 4 times a.

a is 2.

Times 2.

Times c.

c is minus 4.

So times minus 4.

All of that over 2a.

a is 2, so 2 times a is 4.

So that becomes 1 plus or
minus the square root.

So we have a 1.

So we have minus 4 times
a 2 times a minus 4.

That's the same thing as a plus
4 times 2 times a plus 4.

Let's just take that
minus out.

So it's plus.

There's no minus here.

So let's see, 4 times 2 is 8.

Times 4 is 32.

Plus 1 is 33.

All of that over 4.

Let's see, we're not
quite there yet.

Well they say, which is one of
the solutions to the equation?

So let's see.

If we wanted to simplify
this out a-- well,

this is right here.

Because we have 1 plus
or minus the square

root of 33 over 4.

Well they wrote just
one of them.

They wrote just the plus.

So C is one of the solutions.

The other one would have been if
you had a minus sign here.

Anyway, next problem.

56.

And this is another one I
need to cut and paste.

It says, which statement best
explains why there's no real

solution to the quadratic
equation?

OK, so I already have
a guess of why this

won't have a solution.

But in general-- well, let's
try the quadratic equation.

Before even looking
at this problem,

let's get an intuition.

It's negative b plus or minus
you the square root of b

squared minus 4ac, all
of that over 2a.

My question is to you, when does
this not make any sense?

Well you know, this'll work
for any b, any 2a.

But when does the square root
sign really fall apart, at

least when we're dealing
with real numbers,

and that's a clue?

Well, it's when you have a
negative number under here.

If you end up with a negative
number under the square root

sign, at least if we haven't
learned imaginary numbers yet,

you don't know what to do.

There's no real solution to
the quadratic equation.

So if b squared minus
4ac is less than

0, you're in trouble.

There's no real solution.

You can't take a square root of
a negative sign if you're

doing with real numbers.

So that's probably going
to be the problem here.

So let's see what b squared
minus 4ac is.

You have b is 1.

So 1 minus 4 times a.

a is 2.

2 times c is 7.

And sure enough, 1 times 4 times
2 times 7 is going to be

less than 0.

So let's just see what
they have here.

Right, the value of 1
squared-- oh, right.

It's b squared.

Well 1 squared, same
thing as 1.

1 squared minus 4
times 2 times 7,

sure enough is negative.

So that's why we don't
have a real

solution to this equation.

Next problem.

I'm actually out of space.

OK, they want to know
the solution set to

this quadratic equation.

I'll just copy and paste.

So that's essentially the
set of the x's that

satisfy this equation.

And obviously, for any x that
you put in this, the left-hand

side is going to
be equal to 0.

So what x's are valid?

And they just want us to apply
the quadratic equation.

So we've written it a couple of
times, but let's just do it

straight up.

So it's negative b.

b is 2.

So it's negative 2
plus or minus the

square root of b squared.

Well that's 2 squared.

Minus 4 times a.

a is 8.

Times c, which is 1.

All of that over 2 times a.

So 2 times 8, which is equal to
minus 2 plus or minus the

square root of 4-- let's see.

Did I write this down?

Negative b plus or minus the
square root of b squared minus

4 times a times c.

Right.

So you get 4 minus 32.

That's why I was double checking
to see if I did this

right because I'm going to get
a negative number here.

All of that over 16.

And so we're going to end up
with the same conundrum we had

in the last. 4 minus 32, we're
going to end with minus 2 plus

or minus the square root
of minus 28 over 16.

And if we're dealing with real
numbers, I mean there's no

real solution here.

And at first I was worried.

I thought I made a careless
mistake or there was an error

in the problem.

But then I look at
the choices.

They have choice D.

And I'll copy and paste
choice D here.

Choice D.

No real solution.

So that's the answer, because
you can't take a square root

of a negative number and stay
in the set of real numbers.

Let's see, do I have time
for another one?

I'm over the 10 minutes.

I'll wait for the next video.

See you soon.