Question

Which ones are a perfect square trinomials?

9x^2− 6x + 1

36b^2-24b + 8

16x^2+ 24x + 9

4a^2− 10a + 25

Answer 1

Perfect square trinomials are trinomials where
1. The first and last terms are perfect squares.
2. The coefficient of the second term equals double the product of the square-roots of the first and third terms.
I will give an example:
\( x^2+8x+16 \)

1. 1 and 16 are both perfect squares.
The square roots of these coefficients are 1 and 4.
2. Double the product of 1 and 4 gives 8, which matches the coefficient of the second term.
Therefore the given trinomial is a perfect square trinomial.

Answer 2

An additional note:
The middle coefficient could be positive or negative and still be a perfect square trinomial.

Answer 3

But what if the coefficient is not a 1?

Answer 4

\( 25x^2+40x+16\)
25 and 16 are perfects (square roots = 5 and 4)
double the product =2*5*4=40 = coef. of middle term.
=> \( 25x^2+40x+16\) is a perfect square trinomial.

Answer 5

The factorization is : \( 25x^2+40x+16\) = \( (5x+4)^2 \)

Answer 6

If the middle coefficient is negative, the it would be (5x-4)^2

Answer 7

So is the 1st, 2nd, and last one a perfect trinomial?

Answer 8

For the second one, if you factor out the common factor, the last term is 2, which is not a perfect square, so...
By now, you can factorize (by sight) the first, third and fourth.

Answer 9

Ohh ok THANK YOUU @mathmate !!!!!!!!!

Answer 10

You're welcome!
Note: Do remember that the first and last term must be BOTH positive or negative for the trinomial to be a perfect square. If they are both negative, then factor out the negative sign, example:
\( -3x^2+6x-3 \\\)
can be factored as
\( -3(x^2-2x+1) \)
The part in parentheses is a perfect square trinomial, but with the factor, it won't be, because -3 is itself not a perfect square.