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4x2 + 8x + 16
x2 +9x + 9
x2 -6x +36
4x2 - 12x + 9
In order for these trinomials to be perfect squares, their first terms and their last terms
need to be perfect squares and positive. That's not a help because 4x^2, 16, x^2, 9, and 36
are all perfect squares and all are positive. That means all the trinomials could be perfect squares.
Let's look at the second and third trinomials. In the second, the + 9 factors into either
+ 3 times +3 or -3 times -3. But there is no way that either +3x and +3x or -3x and -3x can
be added to give the middle term +9x of the trinomial (+3x and -3x are the products of
the square root of the last term in the trinomial times the square root of the first term
of the trinomial.) Because you can't combine the possible products of the square root of the
last term and the first term to get the middle term, that eliminates the the trinomial from
being a perfect square.
The same thing can be said to happen to the third trinomial. The 36 factors into +6 times +6
or -6 times -6. But there is no way that +6x and +6x can be combined to give the middle term
of -6x. The same can be said of -6x and -6x. They can't be combined to give the middle
term of -6x. Since neither will work, this eliminates the third equation from being a perfect
Now, let's do the first trinomial. Notice that the square root of the first term is
2x. Also note that the square root of +16 is either +4 or -4. Now form the product of
the square roots of these two terms. First the product of 2x and +4 is +8x. Is there any
way that +8x and + 8x can be combined to give the middle term of the trinomial +8x? No.
So how about doing the same thing with the -4 times 2x to get -8x? You can't get -8x and
-8x to add up to +8x either.
So, finally let's try the fourth trinomial of 4x2 - 12x + 9. Again, the square root of the
first term is 2x and the square root of the last term is either +3 or -3. Multiply the
+3 times the 2x and get +6x. Is there anyway that +6x and +6x can be combined to give the
trinomial's middle term of -12x? Nope, but how about the product of -3 and 2x? That product
is -6x. That is -6x. And when you add -6x and -6x you get the middle term of the trinomial.
The factors of the fourth trinomial are both (2x - 3) and (2x - 3). When you multiply
these two together you get back to the original trinomial. Therefore, the answer to
your question is the last of the four trinomials that you were given as potential
Hope this helps you to see how to examine each trinomial to see if it is a perfect