To determine which of the given equations contains a difference of squares, we need to recall that an equation contains a difference of squares if it is of the form [tex]\(a^2 - b^2 = 0\)[/tex], which can be factored as [tex]\((a + b)(a - b) = 0\)[/tex].
Let's analyze each equation step-by-step:
1. Equation 1: [tex]\(4 x^2 - 80 = 0\)[/tex]
We can rewrite this equation as:
[tex]\[
4 x^2 - 80 = 0
\][/tex]
By factoring out the common term:
[tex]\[
4(x^2 - 20) = 0
\][/tex]
The expression inside the parenthesis, [tex]\(x^2 - 20\)[/tex], is not a difference of squares because [tex]\(20\)[/tex] is not a perfect square.
Thus, this equation does not contain a difference of squares.
2. Equation 2: [tex]\(x^2 = 2\)[/tex]
We rewrite this equation as:
[tex]\[
x^2 - 2 = 0
\][/tex]
The expression [tex]\(x^2 - 2\)[/tex] is not a difference of squares because [tex]\(2\)[/tex] is not a perfect square.
Thus, this equation does not contain a difference of squares.
3. Equation 3: [tex]\(9 x^2 = 69\)[/tex]
We rewrite this equation as:
[tex]\[
9 x^2 - 69 = 0
\][/tex]
By factoring out the common term:
[tex]\[
9 (x^2 - \frac{69}{9}) = 0
\][/tex]
Simplify the fraction:
[tex]\[
9 (x^2 - \frac{23}{3}) = 0
\][/tex]
The expression inside the parenthesis, [tex]\(x^2 - \frac{23}{3}\)[/tex], is not a difference of squares because [tex]\(\frac{23}{3}\)[/tex] is not a perfect square.
Thus, this equation does not contain a difference of squares.
4. Equation 4: [tex]\(36 x^2 - 100 = 0\)[/tex]
We rewrite this equation as:
[tex]\[
36 x^2 - 100 = 0
\][/tex]
By factoring or recognizing the structure:
[tex]\[
36 x^2 - 100 = 0
\][/tex]
Recognize [tex]\(36 x^2\)[/tex] and [tex]\(100\)[/tex] are perfect squares:
[tex]\[
(6x)^2 - 10^2 = 0
\][/tex]
This is indeed a difference of squares, and can be factored as:
[tex]\[
(6x + 10)(6x - 10) = 0
\][/tex]
Thus, the equation [tex]\(36 x^2 - 100 = 0\)[/tex] contains a difference of squares. Therefore, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
