In this lesson, we will learn how to factor a perfect square trinomial using a shortcut
factoring formula.
A perfect square trinomial will always fit one of these two forms here.
Where if the form is a squared plus 2ab plus b squared, the trinomial factors into the
square of the quantity a plus b.
And if the perfect square trinomial fits form a squared minus 2ab plus b squared, the perfect
square trinomial factors in to the square of the quantity a minus b.
Notice how if we have plus 2ab the binomial is a plus b.
If we have minus 2ab the binomial is a minus b.
I so want to mention the factoring techniques of grouping as well as trial and error do
still work on perfect square trinomials, but in this lesson we will focus on the short
cut formula to factor perfect square trinomials.
And therefore to factor a perfect square trinomial, step one, we confirm the first term and third
term are perfect squares.
This allows up to identify the values of a and b.
Step two, we confirm the middle term is equals to 2ab or 2 times a times b.
Then step three, we write the trinomial in factored form, which is the quantity a plub
b squared or the quantity a minus b squared based upon the sign of the middle term.
Our first example is 36 x squared plus 60 x plus 25.
The first step is to verify that 36 x squared and 25 are both perfect squares.
36 x squared is a perfect square because it's equal to the square of 6x. 25 is a perfect
square because it is equal to the square of 5.
Written in this form, this shows that a is equal to 6x and b is equal b 5.
The next step is to verify the middle term is equal to 2ab.
2ab is equal to 2 times 6 x times 5, which does equal 60 x, the middle term.
And therefore we do have a perfect square trinomial and the trinomial will factor into
the square of the quantity a plus b.
We know the binomial is a plus b because the middle term is plus 2ab or in this case plus
60 x.
Which means the given trinomial factors into the square of the quantity 6 x plus 5.
So this is the factored form of the given perfect square trinomial, but let's multiply
this out and verify this is correct.
To multiply out the square of the quantity 6 x plus 5, there are no shortcuts, we have
two factors of 6 x plus 5.
When multiplying two binomials, we always have 4 products.
One, two, three, and four.
6 x times 6 x equals 36 x squared.
6x time 5 is 30x, so plus 30x. 5 times 6x also equals 30 x, so plus 30x.
And finally, 5 times 5 equals 25, so we have plus 25.
Simplifying, 30 x plus 30 x equals 60x giving us 36 x squared plus 60 x plus 25, verifying
this is factored correctly.
Let's take a look at our second example.
We have 49 x squared minus 28 x plus 4.
Notice how because the middle term is minus 28 x, if we have a perfect square trinomial,
we'll be using this factoring formula here where we have the square of the quantity a
minus b.
Step one is to verify the first term and the third term are perfect squares.
49 x squared is a perfect square.
It's equal to the square of 7 x.
4 is a perfect square.
It's equal to two squared.
Written in this form we can easily identify the values of a and b.
a equals 7 x.
b equals 2.
The next step is to verify 2ab is the middle term or in this case 2ab equals 28 x.
So 2ab equals 2 times 7x times 2, which does equal 28 x.
So we do have a perfect square trinomial in this form here, which means it factors into
the square of the quantity a minus b, which means the given trinomial factors into the
square of the quantity 7 x minus 2.
This is the factored form of the perfect square trinomial, but once again lets' verify this
by multiplying this out.
The square of the quantity 7 x minus 2 equals two factors of 7 x minus 2.
To multiply we have 4 products, one, two, three, and four.
7 x times 7 x equals 49 x squared. 7 x times negative 2 equals negative 14 x or minus 14
x.
Negative 2 times 7 x equals negative 14 x or minus 14 x.
Negative two times negative 2 is equal to positive 4.
Combining like terms we have 49 x squared minus 28 x plus 4, verifying our factored
form is correct.
I hope you found this helpful.