CA Algebra I: Quadratic Roots

58-62, x-intercepts of a quadratic function

We're on problem 58.

The graph of the equation y is
equal to x squared minus 3x

minus 4 is shown below.

Fair enough.

For what value or values
of x is y equal to 0?

So they're essentially
saying is, when does

this here equal 0?

They want to know when
does y equal 0?

So what values of x
does that happen?

And we could factor this and
solve for the roots, but they

drew us the graph, so let's
just inspect it.

So when does y equal 0?

So let me draw the line
of y is equal to 0.

So that's right here.

Let me draw it as a line.

y equals 0.

That's y equals 0 right there.

So what values of x
makes y equal 0?

If I can see this properly,
it's when x is equal to

negative 1 and when
x is equal to 4.

So x is equal to negative
1 or 4.

And if we substitute either of
these values into this right

here, we should get
y is equal to 0.

And let's see.

The choices, do they have
negative 1 and 4?

Yep, sure enough.

Negative 1 and 4 right there.

Next question, 59.

Let me copy and paste it.

OK, I've copied it.

I'll paste it below.

I'll do it right
on top of this.

There you go.

59.

Let me erase this stuff
right here.

This is unrelated
to this problem.

So they are asking-- let me get
the pen right-- which best

represents the graph of
y is equal to minus x

squared plus 3?

So here, just to get an
intuition of what parabolas

look like, because these are all
parabolas, or the graph of

a quadratic equation.

So if I had the graph of y is
equal to x squared, what does

that look like?

Let me just draw a quick and
dirty x- and y-axis.

And I think you're familiar
with what that looks like.

So if I were to just draw y is
equal to x squared, that looks

something like this.

It looks something like this.

It will look something
like that.

I think you're familiar
with it.

Because you're taking x squared,
you always get

positive values.

Even if you have a negative
number squared that still

becomes a positive.

And it's symmetric around the
line x is equal to 0.

That's the graph of y
equals x squared.

Now let me ask you a question.

What is the graph of y is equal
to minus x squared?

Let me do that.

y is equal to minus x squared.

So it's essentially the same
thing as this graph, but it's

going to be the negative
of whatever you get.

So here, x squared is always
going to be positive.

You square any real number
and you're going to

get a positive number.

Here, you square any real
number, this part right here

becomes positive.

But then you take the
negative of it.

So this is always going to
be a negative number.

So x equals 0 is still going to
be there, but regardless of

whether you go in the positive
x direction or the negative x

direction, this is going
to be positive.

But then you put the
negative sign, it's

going to become negative.

So the graph is going
to look like this.

The graph is going to
look like that.

I didn't draw that well, let
me give that another shot.

The graph is going to
look like that.

It's essentially like the mirror
image of this one if

you were to reflect
it on the x-axis.

This is y equals minus
x squared.

So now you're pointing
it down.

The u goes-- opens
up downward.

And hopefully that makes
a little bit of sense.

And now, what happens if
you do plus or minus 3?

So what is y is equal
to x squared plus 3?

Not minus x squared plus 3,
but just x squared plus 3.

So if you start with x squared,
now every y value for

every given x is just going
to be 3 higher.

So it's just going to shift
the graph up by 3.

It's going to look like that.

So if you go from x squared to
x squared plus 3, you're just

shifting up by 3.

Similar, if you go from minus
x squared to minus x squared

plus 3, which is what they gave
us in the problem, you're

just going to shift
the graph up by 3.

So I'll do that in
this brown color.

So it's just going to take this
graph, which is minus x

squared and you're going
to shift it up by 3.

So it's going to look
something like this.

It's going to look something
like that.

So let's see, out of all the
choices they gave us, it

should be opening downward and
it should have its y-intercept

at y is equal to 3.

If you put x is equal to
0, y is equal to 3.

So let's see.

It's opening downwards.

So these two are the only two
that are opening downwards.

And the y-intercept should
be at 3 because we

shifted it up by 3.

So this is the choice B.

Problem 60.

Which quadratic funtion, when
graphed, has x-intercepts of 4

and minus 3?

So x-intercepts of 4 and minus
3 means that when you

substitute x of either of
these values into the

equation, you get
y is equal to 0.

Because when y is equal to 0,
you're at the x-intercepts.

This is when y equals to 0.

So that's what they mean
by x-intercepts.

So how do we set up an equation
where if I put in one

of these numbers I'm
going to get 0?

Well, if I make it the product
of x minus the first root and

x minus the second root.

So x minus minus
3 is x plus 3.

So think about it.

If you put 4 here for x, you
get 4 minus 4, which is 0.

Times 4 plus 3 is 7.

So 0 times 7 is 0.

So that works.

And then, for minus 3.

Minus 3 minus 4 is minus 7.

But then minus 3
plus 3 is a 0.

So either of these, when you
substitute it into this

expression, you get 0.

Let's see, which
choice is that?

x minus 4 times x plus 3.

Have the x-intercepts
of 4 and minus 3.

Right.

This should be right.

They're being tricky
right here.

So x plus 3 is there.

I see that in a couple
of them, right?

But I don't see the x
minus 4 anywhere.

But that's because we can
multiply this by any constant.

Because 0 times some number
times some constant is still

going to be 0.

So if you look at this one
right here, 2x minus 8.

We could factor out a 2.

That's the same thing as
2 times x minus 4.

So choice B is x plus 3,
times x minus 4, times

some constant 2.

So choice B is our answer.

If this is equal to 0 when x is
equal to 4 minus 3, this--

any constant, x minus 4 times x
plus 3--I that's still going

to be equal to 0.

Because when x is equal
to 4, this is going

to be equal to 0.

So 0 times anything times
anything else

is going to be 0.

Same thing with x is
equal to minus 3.

So they just put a 2 here.

That's a good problem.

It made you realize that you
could put a constant in there

and it's a little tricky.

Next problem.

OK, so they want to know-- let
me copy and paste it-- how

many times does the graph of y
equals 2x squared minus 2x

plus 3 intersect the x-axis?

So the easiest thing-- because
maybe it doesn't intersect the

x-axis at all.

Maybe if you use a quadratic
equation

there are no real solutions.

So let's just apply
the quadratic.

So the roots or the times that--
I guess the x values

that solve this equation, 2x
squared minus 2x plus 3 is

equal to 0.

And these are the x
values where you

intersect the x-axis.

And why do I say that?

Because the x-axis is the
line y is equal to 0.

So I set y equal to
0 and I get this.

And we know from the quadratic
equation the solution to this

is negative b-- let me do
this in another color.

So negative b.

So minus minus 2 is 2.

Plus or minus the square
root of b squared.

Minus 2 squared is 4.

Minus 4 times a, which is 2.

Times c, times 3.

All of that over 2a.

2 times 2, which is 4.

Now they don't want
us to figure out

the roots or anything.

They just want to know,
how many times does it

intersect the axis?

So let's think about this.

What happens under this
radical sign?

We have 4 times 2
times 3 is 24.

So this becomes 2 plus or
minus 4 minus 24 over 4.

This is minus 20.

So you end up with minus 20
under the radical sign.

And we know if we're dealing
with real numbers, if we want

real solutions, you can't take
the square root of minus 20.

So this actually has no
solutions or, another way to

put it is, there is no x values
where y is equal to 0.

Or another way to put it
is, this never does

intersect the x-axis.

So it's A.

none.

What gave that away was the fact
that when you apply the

quadratic equation, you get a
negative number under the

radical sign.

So if we're dealing
with real numbers,

there's no answer there.

Next question.

62.

An object that is projected
straight down-- oh, this is

good, this is projectile
motion-- is projected straight

downward with initial velocity
v feet per second, Travels a

distance of s v times t plus
16t squared, where t equals

time in seconds.

If Ramon is standing on a
balcony 84 feet above the

ground and throws a penny
straight down with an initial

velocity of 10 feet per second,
in how many seconds

will it reach the ground?

OK, so he's 84 feet
above the ground.

Let's draw a diagram.

He's 84 feet.

This is 84 feet above
the ground.

It says, how many seconds will
it reach the ground?

So we essentially want to know
how many seconds will it take

it to travel? s is
distance, right?

So s is equal to 84 feet.

It has to go down 84 feet.

Let's see if we can
figure this out.

Now this is something that might
be a little bit-- so how

long does it take it to go 84
feet, I guess is the best way

to think about it.

So we say 84 is equal to
velocity times-- your initial

velocity times time.

And your initial velocity
is 10 feet per second.

So it's 10 feet per second.

Everything is in feet I think.

Right, everything is--
v feet per second.

Initial velocity of
v feet per second.

So 10 feet per second
times time.

I just substituted what
they gave us.

10 for v.

Plus 16t squared.

And now I just solve
this quadratic.

That is a t right there.

So let me put everything on the
same-- let me subtract 84

from both sides and I'll
rearrange a little bit.

So you get 16t squared plus 10t
minus 84 is equal to 0.

Well I swapped the sides.

I put these on the left.

Well, let me just show
you what I did.

I swapped the sides.

So I made this 16t squared
plus 10t equals 84.

I just swapped them.

And then I subtracted 84 from
both sides to get this.

And now we just have to
solve when t equals 0.

So I guess the first thing we
could do is we could simplify

this a little bit.

Everything here is
divisible by 2.

So this is 8t squared plus-- I'm
just dividing both sides

of this equation by 2.

Plus 5t minus what?

Minus 42 is equal to 0.

And then we can use the
quadratic equation.

So what are the solutions? t
is equal to negatives b.

So minus 5 plus or minus the
square root of b squared, so

25, minus 4 times a.

times 8, times c.
c is minus 42.

So instead of times minus
42, let's put a plus

here and do plus 42.

Just a negative times a negative
is a positive.

All of that, over 2 times a.

2a is 16.

So let's see where
that gets me.

So I t is equal to minus 5 plus
or minus the square root.

What is this?

25 plus-- let's see.

4 times 8 times 42.

That's 32 times 42.

32 times 42.

2 times 32 is 64.

Put a 0.

4 times 2 is 8.

4 times 3 is 12.

You end up with 4,
14, 3 and 1.

So this is 1,344.

And we're going to
add this 25 here.

So let me see.

Plus 1,344.

All of that over 16.

Let's see.

1,344 plus 25.

So it's minus 5 plus or
minus the square root

of-- what is this?

1,369 over 16.

And actually, I don't
know what the square

root of 1369 is.

Let me get the calculator.

Let me open it up.

Give me one second.

Accessories, calculator.

All right.

So 1,369.

37.

Look at that.

OK, so it's minus 5
plus or minus 37.

That's the square
root of 1,369.

So minus 5 plus or minus 37 over
16 is equal to the time.

Now we don't have to worry about
the minus because that's

going to give us a
negative number.

Minus 5 minus 37 over 16.

We don't want a negative time,
we want a positive time.

So let's just do the positive.

So minus 5 plus 37.

Let's see.

Minus 5 plus 37 over 16.

So that's 32/16, which
equals to 2 seconds.

And that's choice A.

Anyway, see you in
the next video.