To determine which of the given trinomials are perfect squares, we need to recognize whether each trinomial can be expressed in the form [tex]\((ax + b)^2 = a^2x^2 + 2abx + b^2\)[/tex].
Let's analyze each trinomial step-by-step.
1. Trinomial: [tex]\(x^2 + 20x + 100\)[/tex]
- This can be expressed as [tex]\((x + 10)^2\)[/tex] because:
[tex]\[
(x + 10)^2 = x^2 + 2 \cdot 10 \cdot x + 10^2 = x^2 + 20x + 100
\][/tex]
Therefore, [tex]\(x^2 + 20x + 100\)[/tex] is a perfect square trinomial.
2. Trinomial: [tex]\(x^2 - 10x + 5\)[/tex]
- This does not match the form [tex]\((ax + b)^2\)[/tex] because the middle term does not equal [tex]\(2ab\)[/tex] (where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are constants and [tex]\( \frac{-10}{2a} = b \)[/tex].
3. Trinomial: [tex]\(x^2 + 6x + 9\)[/tex]
- This can be expressed as [tex]\((x + 3)^2\)[/tex] because:
[tex]\[
(x + 3)^2 = x^2 + 2 \cdot 3 \cdot x + 3^2 = x^2 + 6x + 9
\][/tex]
Therefore, [tex]\(x^2 + 6x + 9\)[/tex] is a perfect square trinomial.
4. Trinomial: [tex]\(x^2 + 4x + 4\)[/tex]
- This can be expressed as [tex]\((x + 2)^2\)[/tex] because:
[tex]\[
(x + 2)^2 = x^2 + 2 \cdot 2 \cdot x + 2^2 = x^2 + 4x + 4
\][/tex]
Therefore, [tex]\(x^2 + 4x + 4\)[/tex] is a perfect square trinomial.
5. Trinomial: [tex]\(x^2 + 6x - 9\)[/tex]
- This doesn't match the form [tex]\((ax + b)^2\)[/tex] for any constants [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
6. Trinomial: [tex]\(x^2 - 8x - 16\)[/tex]
- This doesn't match the form [tex]\((ax + b)^2\)[/tex] for any constants [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
So, the perfect square trinomials from the given list are:
- [tex]\(x^2 + 20x + 100\)[/tex]
- [tex]\(x^2 + 6x + 9\)[/tex]
- [tex]\(x^2 + 4x + 4\)[/tex]
These are the trinomials that can be expressed as the square of a binomial.
