To determine which of the given equations contains a perfect square trinomial, we need to understand what a perfect square trinomial is. A quadratic expression [tex]\(ax^2 + bx + c\)[/tex] is a perfect square trinomial if it can be written in the form [tex]\((ax + b)^2\)[/tex]. A common characteristic of a perfect square trinomial is that the constant term [tex]\( c \)[/tex] is a perfect square, and [tex]\( b \)[/tex] is twice the product of [tex]\( a \)[/tex] and the square root of [tex]\( c \)[/tex].
Let's analyze each equation step-by-step:
1. Equation 1: [tex]\(x^2 + 14x + 49 = 0\)[/tex]
We rewrite it as:
[tex]\[
x^2 + 14x + 49
\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = 14\)[/tex], and [tex]\(c = 49\)[/tex].
We check if:
[tex]\[
b^2 = 4ac
\][/tex]
[tex]\[
14^2 = 4 \cdot 1 \cdot 49
\][/tex]
[tex]\[
196 = 196
\][/tex]
This equation is a perfect square trinomial because it can be factored as:
[tex]\[
(x + 7)^2 = 0
\][/tex]
2. Equation 2: [tex]\(x^2 - 5x + 64 = 0\)[/tex]
We rewrite it as:
[tex]\[
x^2 - 5x + 64
\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = -5\)[/tex], and [tex]\(c = 64\)[/tex].
We check if:
[tex]\[
b^2 = 4ac
\][/tex]
[tex]\[
(-5)^2 = 4 \cdot 1 \cdot 64
\][/tex]
[tex]\[
25 \neq 256
\][/tex]
This equation is not a perfect square trinomial.
3. Equation 3: [tex]\(x^2 - 6x + 72 = 0\)[/tex]
We rewrite it as:
[tex]\[
x^2 - 6x + 72
\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = -6\)[/tex], and [tex]\(c = 72\)[/tex].
We check if:
[tex]\[
b^2 = 4ac
\][/tex]
[tex]\[
(-6)^2 = 4 \cdot 1 \cdot 72
\][/tex]
[tex]\[
36 \neq 288
\][/tex]
This equation is not a perfect square trinomial.
4. Equation 4: [tex]\(x^2 + 2x - 4 = 0\)[/tex]
We rewrite it as:
[tex]\[
x^2 + 2x - 4
\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(c = -4\)[/tex].
We check if:
[tex]\[
b^2 = 4ac
\][/tex]
[tex]\[
2^2 = 4 \cdot 1 \cdot (-4)
\][/tex]
[tex]\[
4 \neq -16
\][/tex]
This equation is not a perfect square trinomial.
Based on these calculations, we find that only the first equation:
[tex]\[
x^2 + 14x + 49 = 0
\][/tex]
contains a perfect square trinomial.
So, the answer is the first equation.
