We're asked to find the value
that completes the square,
then write the trinomial as a perfect square.
We've learned from our lesson that if we have a trinomial
with a leading coefficient of one that's a perfect square,
it must fit the form shown here
where the constant term C
must be equal to B divided by two squared,
where B is equal to the coefficient of the degree one
or liner term, or in this case, the coefficient of X.
So notice in our first example,
the coefficient of X is positive 3/2,
so we'd have B equals 3/2
and on this time, because B is equal to a fraction,
instead of using the formula
C equals B divided by two squared,
instead of dividing by two, let's multiply by 1/2
and write this as 1/2 times B squared,
which is equivalent, but again, when B is a fraction,
this form of the formula for C is going to be more helpful.
So notice here, we would have C equals,
we'd have 1/2 times 3/2
squared, which would give us 3/4 squared,
which is equal to 9/16,
which means if you add 9/16,
we now have a perfect square trinomial
which we now want to factor,
which will factor into two binomial factors
that are equal to each other.
We'll begin with the first terms in the binomial factors
which must be the factors of X squared which are X and X.
Now typically we ask what are the factors of 9/16
that add to 3/2 or the factors of the constant term
that add to the coefficient of X,
but that's not an easy question here
because of the fractions
and that's why it's important to show
the work I've shown here to determine the value of C.
Notice how we squared 3/4 to obtain 9/16,
which means the factor of 3/4
is going to be the constant term
in each binomial factor.
One factor is going to be X plus 3/4
and the second factor is also going to be X plus 3/4,
which means we can write this as the quantity
X plus 3/4 squared.
Let's also verify this is factored properly
by verifying that 3/4 times 3/4 does equal C.
Well, notice how this does equal 9/16, which is C.
Let's also verify that 3/4 plus 3/4
does equal B, which is 3/2.
Notice how here we have 6/4, which does equal 3/2,
verifying this is factored properly
and now for our second example,
notice how the coefficient of the degree one term
where the coefficient of X is negative 4/5,
so B equals negative 4/5
and that means that the constant C
must be equal to 1/2 times negative 4/5
squared in order to form a perfect square trinomial.
We'll notice how we can simplify here before multiplying.
There's one factor of two and two
and two factors of two and negative four,
so we have, looks like negative 2/5 squared,
which would equal 4/25,
so by adding 4/25, we now have a perfect square trinomial,
which when we write as a perfect square,
many we want to factor.
So then we would have X and X
and again, we typically ask what are the factors of 4/25
that add to negative 4/5, not an easy question,
and again, looking at our work below,
notice how we squared negative 2/5 to obtain 4/25,
which means our binomial factors are X minus 2/5
and X minus 2/5,
which is the quantity X minus 2/5
squared, and again, just to verify this is factored properly
let's verify that negative 2/5
times negative 2/5 is equal to C, or 4/25
and notice how this checks,
which again, should be obvious
because we squared negative 2/5 to obtain 4/25,
but let's also verify that negative 2/5
plus negative 2/5 is equal to B, or negative 4/5
and notice how it is equal to negative 4/5,
so this is factored properly
and again, just to make sure we understand
what is happening here,
we were asked to determine the value
that completes the square,
so here the value was 9/16 that completed the square
and here the value was 4/25
and then as a perfect square, our first example was
the quantity X plus 3/4 squared
and our second example was the quantity X minus 2/5 squared.
Okay, I hope you found these examples helpful.