To determine whether each given expression is a perfect-square trinomial, we need to factor them and see if they can be written as the square of a binomial, i.e., in the form [tex]\((a \pm b)^2\)[/tex].
A perfect-square trinomial has the form:
[tex]\[ a^2 \pm 2ab + b^2 \][/tex]
which can be factored into:
[tex]\[ (a \pm b)^2 \][/tex]
Let's analyze each expression step-by-step.
1. Expression: [tex]\(x^2 - 16x - 64\)[/tex]
For this expression to be a perfect-square trinomial, it must fit the form [tex]\((x \pm b)^2\)[/tex].
First, let's rewrite the general form of a perfect-square trinomial:
[tex]\[ x^2 \pm 2bx + b^2 \][/tex]
Comparing with [tex]\(x^2 - 16x - 64\)[/tex], this doesn't match the form [tex]\(x^2 \pm 2bx + b^2\)[/tex] because the constant term should be positive and 16 should be twice of a certain value [tex]\(b\)[/tex].
Therefore, [tex]\(x^2 - 16x - 64\)[/tex] is not a perfect-square trinomial.
2. Expression: [tex]\(4x^2 + 12x + 9\)[/tex]
Let's factor this expression.
Notice that [tex]\(4x^2\)[/tex] is a perfect square [tex]\((2x)^2\)[/tex], and [tex]\(9\)[/tex] is a perfect square [tex]\((3)^2\)[/tex].
Rewrite the middle term:
[tex]\[ 4x^2 + 12x + 9 = (2x)^2 + 2 \cdot (2x) \cdot 3 + 3^2 \][/tex]
Therefore, it fits the form of a perfect-square trinomial:
[tex]\[ (2x + 3)^2 \][/tex]
Thus, [tex]\(4x^2 + 12x + 9\)[/tex] is a perfect-square trinomial.
3. Expression: [tex]\(x^2 + 20x + 100\)[/tex]
Let's factor this expression.
Notice that [tex]\(x^2\)[/tex] is a perfect square [tex]\((x)^2\)[/tex], and [tex]\(100\)[/tex] is a perfect square [tex]\((10)^2\)[/tex].
Rewrite the middle term:
[tex]\[ x^2 + 20x + 100 = x^2 + 2 \cdot 10 \cdot x + 10^2 \][/tex]
Therefore, it fits the form of a perfect-square trinomial:
[tex]\[ (x + 10)^2 \][/tex]
Thus, [tex]\(x^2 + 20x + 100\)[/tex] is a perfect-square trinomial.
4. Expression: [tex]\(x^2 + 4x + 16\)[/tex]
Let's factor this expression.
Notice that [tex]\(x^2\)[/tex] is a perfect square [tex]\((x)^2\)[/tex], and [tex]\(16\)[/tex] is a perfect square [tex]\((4)^2\)[/tex].
Rewrite the middle term:
[tex]\[ x^2 + 4x + 16 \text { does not match } (x \pm 4)^2 \][/tex]
A careful re-check shows it doesn't fit the (x) validation:
(x+ 4)^2= x^2+8x-16
[tex]\(x^2+ 4x+16\)[/tex] is not a perfect-square trinomial.
In conclusion, the expressions that are perfect-square trinomials are:
- [tex]\(4x^2 + 12x + 9\)[/tex]
- [tex]\(x^2 + 20x + 100\)[/tex]
