To determine which expressions are perfect square trinomials, let's review what a perfect square trinomial is.

A perfect square trinomial is an expression in the form: $a^2 \pm 2ab + b^2$ which factors as: $(a \pm b)^2$

Checking Each Option

1. $x^2 - 9$: This expression has only two terms, and it is a difference of squares, not a perfect square trinomial. So, this is not a perfect square trinomial.

2. $x^2 - 100$: Similarly, this is also a difference of squares with only two terms. Therefore, this is not a perfect square trinomial.

3. $x^2 - 4x + 4$: This is a trinomial, and it can be factored as: $x^2 - 4x + 4 = (x - 2)^2$ So, this is a perfect square trinomial.

4. $x^2 + 10x + 25$: This is also a trinomial, and it can be factored as: $x^2 + 10x + 25 = (x + 5)^2$ Thus, this is a perfect square trinomial.

5. $x^2 + 15x + 36$: This trinomial cannot be factored into the form of a perfect square. It factors as: $x^2 + 15x + 36 = (x + 3)(x + 12)$ Therefore, this is not a perfect square trinomial.

Conclusion

The correct answers are:

• $x^2 - 4x + 4$
• $x^2 + 10x + 25$

Would you like further explanation on factoring or have any questions?

Related Questions:

1. What defines a perfect square trinomial in general terms?
2. How can we quickly identify a difference of squares?
3. Why can some trinomials not be factored into perfect squares?
4. How do perfect square trinomials relate to completing the square?
5. What are some examples of real-world applications of perfect square trinomials?

Tip: A quick way to check if a trinomial is a perfect square is to see if the middle term is twice the product of the square roots of the first and last terms.