Final answer:
Perfect square trinomials are formed by squaring a binomial equation. The constant in the binomial equation becomes the b value in a typical trinomial equation x² + 2ab + b². The numbers in the parentheses of the examples given are used to create the expanded trinomial form.
Step-by-step explanation:
A perfect square trinomial is a trinomial that is the square of a binomial, which means it derives from squaring a binomial equation. For the given perfect square trinomials, the values in the boxes are what the binomials are squared by, namely (x + 3)^2, (x - 3)^2, (x + 6)^2, (x - 6)^2.
The term inside the parentheses is squared, which gives us a pattern of x² + 2ab + b², where x is the variable, and 2ab represents the twice the product of the binomial terms, and b² represents the square of the second term of the binomial.
For instance, if we expand (x + 3)^2, we get x² + 2*3*x + 3², which simplifies to x² + 6x + 9.
Similar logic applies to the other cases: (x - 3)^2 expands to x² - 6x + 9, (x + 6)^2 expands to x² + 12x + 36, and (x - 6)^2 expands to x² - 12x + 36.
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