Final answer:
A perfect square trinomial has the form a² + 2ab + b², derived from squaring a binomial, and is factorable into a binomial squared, useful in solving quadratic equations.
Step-by-step explanation:
The characteristics of a perfect square trinomial are that it can be expressed as the square of a binomial. To determine if a trinomial is a perfect square, it must have the form a² + 2ab + b² where the first and last terms are perfect squares themselves and the middle term is twice the product of the square roots of the first and last terms.
A perfect square trinomial is derived from the process of squaring a binomial, which in algebraic form would be (x + y)² = x² + 2xy + y². Every term in the trinomial will have the same variable with exponents that make them perfect squares when related to the binomial factors.
Moreover, a perfect square trinomial will always factor into a binomial squared. This is particularly useful when solving quadratic equations or in the process called completing the square, which can be applied to convert a quadratic expression into a perfect square trinomial, thereby facilitating the solution of quadratic equations.