Let's begin by writing the formulas to factor a perfect square trinomial.
a^2 + 2 a b + b^2 &= ( a+ b)^2
a^2 - 2 a b + b^2 &= ( a- b)^2Now we can point out some important facts in the left-hand side expressions.
Since sqrt(a^2)=± a and sqrt(b^2)=± b, we can rewrite the left-hand side as follows.
In consequence, a trinomial is a perfect square trinomial if one of the terms, ignoring the sign, is equal to twice the product of the square roots of the other two terms, and these two terms are positive.
Extra
Example
Let's consider the trinomial 5x^2 + 2sqrt(5)x + 1. At first look, it does not look like a perfect square. However, we can write the first term as follows.
5x^2 = (sqrt(5))^2x^2 = (sqrt(5)x)^2
In consequence, the binomial will looks as follows.
(sqrt(5)x)^2 + 2sqrt(5)x + 1
=
( sqrt(5)x)^2 + 2* sqrt(5)x* 1 + 1^2
As we can see, the middle term is twice the product of the square roots of the first and last terms. The trinomial is a perfect square trinomial.
(sqrt(5)x)^2 + 2* sqrt(5)x+ 1 = (sqrt(5)x+1)^2
