Question

State whether each product will result in a perfect-square trinomial.

1. (x + 5) (x + 5)
2. (x + 2) (x- 2)
3. (5x-6) (5x-6)

Answer 1

Expressions (x + 5)(x + 5) and (5x - 6)(5x - 6) result in perfect-square trinomials, while (x + 2)(x - 2) is not a perfect-square trinomial, but rather a difference of squares.

Step-by-step explanation:

The question asks us to determine if the products of given expressions will result in a perfect square trinomial. A perfect-square trinomial is one that can be written as (a+b)2 = a2 + 2ab + b2, where a and b are real numbers.

1. For the expression (x + 5) multiplied by itself, we use the binomial theorem to expand it: (x + 5)(x + 5) = x2 + 10x + 25. This is indeed a perfect-square trinomial, resulting from squaring the binomial (x + 5).
2. The expression (x + 2)(x - 2) is an example of the difference of squares and expands to x2 - 4, which is not a trinomial, and thus not a perfect-square trinomial.
3. Similarly, for (5x - 6)(5x - 6), we expand it using the binomial theorem: (5x - 6)(5x - 6) = (5x)2 - 2×5x×6 + 36, which simplifies to 25x2 - 60x + 36, another perfect-square trinomial.