First, the maximum power of the variable must be 2. The first analysis is done for the coefficient of the term in x^2 and the independent term; the square root of these numbers must be accurate. Then, the product of the square roots above is performed, and the result is multiplied by 2; the result of this calculation must be equal to the coefficient of the term in x.
For the example given we have:
The coefficient of the term in x^2 is 9 → √9 = 3
The independent term is 4 → √4 = 2
So, these numbers have exact square roots. Multiplying these square roots by 2, we have:
2(3)(2) = 12, that is, the coefficient of the term in x.
If the trinomial, meet these conditions, then it is a perfect square trinomial.
Well, you could factor it like normal. If you get the same parenthesis for an answer, then it is a perfect square.
For a shorter method, following the formula ax*2 + bx + c, consider:
is b^2 - 4ac = 0
If yes, then it is a perfect square. If not, then no.
For instance, here, we have a = 9, b = -12, and c = 4. So, plugging in the numbers:
(-12)^2 - 4*9*4 = 0
144 - 144 = 0
0 = 0
So, it would be a perfect square.
As far as what the perfect square would be, that's something else.