Welcome to a lesson
on forming perfect square trinomials.
Perfect square trinomials are trinomials
that factor into two equal binomial factors.
Let's begin by considering trinomials in the form,
x squared plus bx plus c.
Notice in this form, the coefficient of the degree one term,
our linear term, is b,
and the constant term is c.
So here are five examples of perfect square trinomials.
Notice how these trinomials in factored form
are equal to the product of two equal binomial factors,
which we can write as a binomial squared.
And again, this is the reason why these are considered
perfect square trinomials.
The main goal of this lesson
is to determine what the value of c would have to be,
or the value of the constant term would have to be
in order to form a perfect square trinomial.
To do this, we want to develop a relationship
between the value of b and the value of c.
One way to approach this would be to consider
the product of the quantity x plus d squared.
Well, we know the quantity x plus d squared
would be equal to the product
of the quantity x plus d times itself,
and now to multiply this out, we'd have four products.
We would first distribute the x,
so we'd have x times x, which is x squared,
plus x times d, which would be dx.
And then we'd distribute the d into the second binomial,
so we'd have plus d times x, or plus dx.
And finally, plus d times d or plus d squared.
Now, notice how we have two like terms here,
dx plus dx equals two dx,
which means x squared plus two dx plus d squared
is a perfect square trinomial.
And now, in this form,
we should be able to recognize a relationship
between the constant term d squared,
and the coefficient of the degree one term,
or x term, which is two d.
Notice how the constant term d squared
would be equal to the square of half of the coefficient
of the degree one or linear term.
So more precisely, in this case,
notice that d squared, the constant term,
is equal to, well half of the coefficient of x
would be two d divided by two.
And then if we square this,
we do get the constant term, d squared.
Notice this simplifies to d inside the parentheses,
so we do get d squared equals d squared,
which means if we have a trinomial
in the form x squared plus bx plus c,
the constant term c would have to be equal to
half of b squared.
Notice here b is the coefficient
of the degree one or a linear term.
So if I go back to our five examples,
in each of these five trinomials,
the constant term c is equal to
b divided by two squared.
Well, if I wanted to, instead of dividing by two,
we could multiply by 1/2
and write it as c equals 1/2 times b squared.
Let's go ahead and identify the values of b and c
in each of our trinomials.
Notice here we have b equals 10.
Here we have b equals negative eight,
b equals negative six,
b equals three, and b equals negative 1/3.
Notice here we have c equals 25,
c equals 16, c equals nine,
c equals 9/4, and c equals 1/36.
Let's just check a couple of these
to verify that the constant term c
equals b divided by two squared.
So notice in our first trinomial,
we have b equals 10.
Notice that 10 divided by two squared
would be five squared at 25, which is the value of c.
Let's go ahead and jump down to this fourth example
where we have an odd value for b.
Notice here we have equals three,
and notice that three divided by two squared
does give us 9/4, the value of c.
It's also helpful to recognize that when writing it
in factored form, the value that we have here
before we square it ends up being the constant
in each binomial factor.
Let's also look at this last example.
Notice how b is equal to negative 1/3,
so because b is a fraction,
we'll go ahead and use the formula for c in this form,
where we have 1/2
times b, negative 1/3
squared, which should actually be, what?
Negative 1/6 squared,
which equals positive 1/36.
So notice c is equal to positive 1/36,
and notice how the value we squared
to get 1/36 was negative 1/6,
and notice how both binomial factors
are x minus 1/6 and x minus 1/6.
So again, showing this work is going to be helpful
to write the trinomial in factored form.
Now, before we look at some additional examples, though,
I do want to point out why it's so important
that we know how to complete the square
or find this value of c to form a perfect square trinomial.
One reason is when we start solving quadratic equations,
if we have a quadratic equation in this form here,
well, notice how this would be a perfect square trinomial.
It makes it much easier to solve the equation
using square roots, and also more generally,
when we have a quadratic equation in this form here,
we can solve this using what's called the quadratic formula.
If it wasn't for being able to complete the square
on trinomials, we would not be able to form
this special formula, the quadratic formula.
And then finally, perfect square trinomials are helpful
when graphing quadratic equations on the coordinate plane.
If we have a quadratic equation in this form here,
again, notice how we have a perfect square trinomial here.
Then we can easily identify key components to our graph,
which we'll learn more about in the near future.
But notice how by inspecting the equation,
we can easily find the vertex, two comma four,
and because a is negative, the parabola opens down.
So there are a lot of reasons why it's important
to understand how to form perfect square trinomials.
Now let's look at some examples
of forming perfect square trinomials.
We want to find the value that completes the square.
We need to determine the value of the constant term here
that would make this a perfect square trinomial,
then we're asked to write the trinomial in factored form
or as a perfect square.
Looking at the given information,
notice how the coefficient of the degree one term
or linear term is six,
which means we know b equals six.
And now we know that the constant term c
must equal b divided by two squared.
So notice b is six,
so we'd have six divided by two squared,
which would be three squared, which equals nine.
So by adding nine,
we now have formed a perfect square trinomial,
which we now want to factor.
And we can factor this like we normally would
when we have a leading coefficient of one.
So we'd have two binomial factors.
The first terms would be the factors of x squared,
which are x and x.
Now we want the factors of nine that add to six,
which are three and three,
so we'd have plus three and plus three,
which equals the quantity x plus three squared.
But also looking at the work that we showed
to find the value of c,
notice how we squared positive three
to obtain nine,
so the value here that we squared to get c
is also the same constant in each binomial factor.
So we could also use the information here
to write the trinomial in factored form.
And this will helpful when these get more complicated.
Looking at our second example, notice how here we have
b equals negative eight.
And therefore, c
is going to be equal to negative eight
divided by two squared,
which would be negative four squared which equals 16.
So by adding 16 here,
we now have a perfect square trinomial
and now write this in factored form.
So we have x and x.
And here we're looking for the factors of positive 16
that now add to negative eight,
which should be negative four times negative four,
which means we have x minus four, and x minus four,
which we can write as the quantity, x minus four squared.
And again, notice how, to obtain 16,
we squared negative four, which is the constant term
in each binomial factor.
So again, we can use information here
to check that we factored this properly.
Now, let's finish by taking a look
at two examples that are a little more challenging.
Notice in this example,
the coefficient of the degree one term, our linear term,
would be negative one, so we have b equals negative one,
which means c is equal to,
b divided by two squared would be negative one
divided by two squared.
Well, negative 1/2 squared would be 1/4.
So to form a perfect square trinomial,
we add 1/4 here.
And now to factor this trinomial in the more traditional way
is a little more challenging
because we have a fraction for the value of c.
Well, we know the first terms are going to be x and x.
And that's where this work here can be extremely helpful.
Notice how we squared negative 1/2 to obtain 1/4,
which means in our binomial factors
we're going to have x minus 1/2 and x minus 1/2,
which you can write as a quantity, x minus 1/2 squared.
Notice how negative 1/2 times negative 1/2
is equal to positive 1/4, the value of c,
and negative 1/2 plus negative 1/2
is equal to negative one,
which is the coefficient of x,
which verifies this is factored correctly.
And for our last example, notice how we have b equals 2/3,
which means c is going to be equal to, in this case
because b is a fraction,
instead of writing b divided by two,
let's write this as 1/2 times b
or 1/2 times 2/3 squared.
Remember, dividing by two is equivalent
to multiplying by 1/2.
Notice here we can simplify before multiplying,
so we actually have c equals 1/3 squared,
which would be 1/9.
So by adding 1/9, we form a perfect square trinomial.
And again, in factored form,
it's going to be helpful to have this work below.
We'll have x and x.
Notice how we squared 1/3 to get 1/9,
and therefore we'll have plus 1/3 here
and plus 1/3 here.
To check this, notice how 1/3 times 1/3
does equal c, which is 1/9,
and 1/3 plus 1/3
does equal 2/3, the value of b.
So we can go ahead and write this out
as the quantity x plus 1/3 squared.
Okay, I hope you found this helpful.