Hey, guys.
Welcome to Algebra 1.
Today we're going to focus
on finding zeroes
of quadratic functions.
Your knowledge of factoring
and the coordinate plane
and x-intercepts will come
in handy during this lesson.
You ready to get started?
Let's go.

(female narrator)
Example 1: Find the zeroes
of a quadradic function.
y equals x squared
plus x minus 20.
Taking a look
at this first example,
you're asked to find the zeroes
of this quadratic function.
Just keep in mind
that when finding a zero,
you're being asked
to find an x-intercept.
Here you're actually asked
to find the zeroes, plural,
because typically
in our quadratic functions,
you have two x-intercepts.
There will be some times
where you don't,
but for the most part,
with problems in this lesson,
we're going
to have two zeroes.
We know,
to find an x-intercept,
that y is always 0,
right?
X-intercepts
always take the form
of x is some numerical value,
but y is 0.
When given
this quadratic function,
in order to find
the x-intercepts,
we're going to want to start
by replacing y with 0.
So, 0 equals x squared
plus x minus 20.
Okay?
Now we have
this quadratic equation.
Knowing what you do about
solving quadratic equations,
you know that to solve this,
we start by factoring
the right side.
We'll need to find
two numbers that multiply
to give us negative 20,
but combine
to give us positive 1.
For a refresher
on factoring,
check out that video.
That will take you through
the steps to factor quadratics.
For this lesson, I'll assume
you know the ins and outs.
We'll just dive in and
get this right side factored.
The pair of numbers
with a product of negative 20
but a sum of positive 1
are positive 5
and negative 4,
because 5 times negative 4
is negative 20,
and 5 plus negative 4
is positive 1.
Okay?
I'm going to rewrite
that right side
by a product
of its factors:
x plus 5 and x minus 4.
Now to get the zeroes,
or to get the x-intercepts,
I need to solve these two
mini equations.
I'll set both parts equal
to 0, individually,
and see what value of x
in this first group
would give this whole group
a value of 0,
and what value of x
would give the second group
a value of 0.
Here we go:
x plus 5 equals 0.
I'm going to solve that.
And I'll solve
x minus 4 equals 0.
To solve the first one,
subtract 5
from both sides,
so x equals negative 5.
To solve the second one,
add 4 to each side,
so x equals positive 4.
I can tell on
this quadratic function
that my zeroes are located
at negative 5 and 4,
or if I wanted to write them
as ordered pairs,

(negative 5, 0) and (4, 0).
There you go--
you've got your zeroes
for this quadratic function.
Okay? Let's try another one.
Here we're asked to find
the zeroes
of the quadratic function,
y equals x squared
minus 3x minus 10.
Okay?
Again, when you're asked
to find the zeroes,
you're really being asked
to find the x-intercepts.
Those are
interchangeable terms.
For my x-intercept,
I always like to write myself
a little hint that the x value
is some numerical value,
but y must be 0.
To find the zeroes
of this function,
I'm going to start out
by replacing that y with a 0.
So, I'll have
0 equals x squared
minus 3x minus 10.
Then I'm going to get
this right side factored,
to solve this quadratic.
I'm looking
for two numbers
that have a product
of negative 10,
but when added together,
they get negative 3.
For this case
it's going to be
negative 5 and positive 2.
Because negative 5 times
positive 2 is negative 10,
and negative 5
plus positive 2--
or negative 5 plus 2--
is negative 3.
Okay?
Let's rewrite that right side:

(x minus 5) and (x plus 2).
And then let's solve
these two--
what I like to call
mini equations--
two one-step equations,
to get exactly the value
of each of those x-intercepts.
We'll have x minus 5
equals 0,
and we'll have
x plus 2 equals 0.
So we'll add 5
to each side here.
Okay?
So, that'll wipe out,
And x equals 5
is one of my zeroes.
Then I'll subtract 2,
subtract 2 over here.
And x equals negative 2
is my other zero.
Now if I want,
I could stop here.
This is an acceptable answer.
My zeroes are at x equals 5
and negative 2.
Or I could also write these
as ordered pairs.
I could represent them
as (5, 0) and--
let's get
a little more room here--

(negative 2, 0).
Either way is fine.
Okay. All right?
I do believe
it's time for you to try one.
Press pause.
Take a few minutes.
And find the zeroes
of this quadratic function.
To compare your answers
against mine, press play.

(female narrator)
Find the zeroes of
the quadratic function:
y equals x squared
minus 7x plus 12.
All right,
let's see how you did.
First, if I'm being asked
to find the zeroes,
I know I'm being asked to find
the x-intercepts.
And I know those always
take the form
of x as some number,
y is 0.
So to find the zeroes
of this quadratic,
I'll start out
by replacing y with 0.
So 0 equals x squared
minus 7x plus 12.
Now I have
a quadratic equation,
and in order to solve it,
what I'm going to do
is get
that right side factored.
I need two numbers that have
a product of positive 12
but a sum of negative 7.
Going through the factors,
you'll find that it is
negative 4 and negative 3.
Because negative 4
times negative 3 is positive 12,
and negative 4 plus negative 3
is negative 7.
Let's rewrite this right side
as a product of factors.

(x minus 4), (x minus 3).
Okay? Let's get a little more
workspace here.
Now it's time to solve
what I like to call
the mini equations,
so we can see what the value
of these x-intercepts are.
I'll set x minus 4
equal to 0,
and I'll set x minus 3
equal to 0.
Okay?
To solve this one,
plus 4 plus 4,
so x equals 4.
To solve the other,
plus 3 plus 3,
x equals 3.
So there are your x-intercepts.
You have one at 4
and one at 3.
Or, if you wrote them
as ordered pairs,
you have one
located at (4, 0),
and another
located at (3, 0).
All right?
Good job on that!
Now let's take a look
at the graphs
of some quadratic functions,
to make sure that you
understand that connection
between finding the zeroes
algebraically,
when you're given
an algebraic representation
of a quadratic function,
as well as finding the zeroes
when you're given
a graphical representation
of a quadratic function.
Now let's get this
locked in place
so it doesn't move on us.
Again, if you're being asked
to find the zeroes,
what are you actually
being asked to find?
You're really being asked
to find those x-intercepts.
In the case of a graph,
just zone right in
to the x-axis
and see where your parabola
crosses the x-axis.
Here we have two places
really close to each other.
We have one here and we have
another right there.
So we have one zero
located at (negative 2, 0).
I'm going to write that
as an ordered pair.
And then we have another zero
located at (negative 3, 0).
There are our zeroes
for this quadratic function.
Okay?
Let's try another one.
Let me get this
locked in place first.
Switch to the eraser--
there we go.
Now let's get this
locked in place
so it doesn't move on us
when we start analyzing it.
We're being asked
to find our zeroes.
We're actually being asked
to find the x-intercepts,
Zone right in to the x-axis
and see where your parabola
crosses it.
You have two places here--
I've got the highlighter.
We have one zero right here--
there's one x-intercept--
and another right there.
If I wrote them
as ordered pairs,
I'd say I have one zero
at (negative 1, 0)--
that's the location
of that point--
and then I have another zero
located at 1, 2, 3, 4...0,
because that's the location
of that one.
There are my two zeroes
for this quadratic function.
All right?
Now you give this one a try.
Press pause,
take a minute,
and locate the zeroes
of this quadratic function.
To compare your answer
against mine, press play.

(female narrator)
The bottom of the parabola
is just under the x-axis
and crosses the x-axis
at 1 and 2.
All right?
Let's see how you did.
Let me get my graph
locked in place here
so it doesn't start
to move on me.
Let's locate these zeroes.
If I'm asked
to find the zeroes,
I'm really
finding the x-intercepts.
And I have one
located right here,
and another right there.
I'm just going to write those
locations as ordered pairs.
So, I have (1, 0)
right here at 1, 0,
believe it or not.
And I have another zero
located right here at (2, 0).
There are the locations
of my two zeroes
for my quadratic function.
All right,
great job, guys,
finding the zeroes
of your quadratic functions.
I hope you saw how factoring
and your knowledge
of the coordinate plane
came in handy for this lesson.
Hope to see you back here soon
for more Algebra 1. Bye!