To determine which of the given trinomials are perfect square trinomials, we need to analyze each one individually. A perfect square trinomial is one that can be written in the form [tex]\((ax + b)^2\)[/tex], resulting in the expanded form [tex]\(a^2x^2 + 2abx + b^2\)[/tex].
Let's check each trinomial:
1. [tex]\(x^2 - 9\)[/tex]:
- This can be rewritten as [tex]\(x^2 - 3^2\)[/tex].
- It is a difference of squares, which factors as [tex]\((x - 3)(x + 3)\)[/tex], not a perfect square trinomial.
2. [tex]\(x^2 - 100\)[/tex]:
- This can be rewritten as [tex]\(x^2 - 10^2\)[/tex].
- Similar to the previous case, it is a difference of squares: [tex]\((x - 10)(x + 10)\)[/tex], not a perfect square trinomial.
3. [tex]\(x^2 - 4x + 4\)[/tex]:
- This trinomial can be rewritten as [tex]\((x - 2)^2\)[/tex].
- Expanding [tex]\((x - 2)^2\)[/tex] gives [tex]\(x^2 - 4x + 4\)[/tex], which matches the original trinomial. It is a perfect square trinomial.
4. [tex]\(x^2 + 10x + 25\)[/tex]:
- This trinomial can be rewritten as [tex]\((x + 5)^2\)[/tex].
- Expanding [tex]\((x + 5)^2\)[/tex] gives [tex]\(x^2 + 10x + 25\)[/tex], which matches the original trinomial. It is a perfect square trinomial.
5. [tex]\(x^2 + 15x + 36\)[/tex]:
- We attempt to factor it, but it does not match the form [tex]\((x + a)^2\)[/tex] or [tex]\((x - a)^2\)[/tex].
From the analysis, the trinomials which turn out to be perfect square trinomials are:
- [tex]\(x^2 - 4x + 4\)[/tex]
- [tex]\(x^2 + 10x + 25\)[/tex]
Thus, the perfect square trinomials are:
- [tex]\(x^2 - 4x + 4\)[/tex]
- [tex]\(x^2 + 10x + 25\)[/tex]
