To determine the characteristics of quadratic equations that contain a difference of squares, let's begin by understanding the general form of such equations.
A quadratic equation typically has the form:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
However, a quadratic equation that contains a difference of squares can be written in a different specific form, which is:
[tex]\[ a(x^2) - c = 0 \][/tex]
This form can be derived by recognizing the structure of the difference of squares:
[tex]\[ (x^2 - a^2) = (x + a)(x - a) \][/tex]
But in a more general term considering the equation [tex]\( a(x^2) - c = 0 \)[/tex], we note the following:
1. There is no [tex]\(bx\)[/tex] term, which implies that [tex]\( b = 0 \)[/tex].
Based on this structure:
[tex]\[ a(x^2) - c = 0 \][/tex]
The middle term ([tex]\(bx\)[/tex]) is missing; hence, [tex]\( b \)[/tex] has to be zero ([tex]\( b = 0 \)[/tex]).
This tells us something special about the value of [tex]\( b \)[/tex] in such quadratic equations:
- The value [tex]\( b \)[/tex] must be [tex]\( 0 \)[/tex].
Now, reviewing the given options:
1. The value [tex]\(|b| = 2 \sqrt{a} \sqrt{c}\)[/tex]: This statement is not true given our structure. The value of [tex]\( b \)[/tex] in the standard form is derived to be 0, irrespective of the values of [tex]\( a \)[/tex] and [tex]\( c \)[/tex].
2. Only the value of [tex]\( c \)[/tex] is a perfect square: This is not necessarily true. While [tex]\( c \)[/tex] could be a perfect square, it is not a necessary condition for all quadratic equations of this type.
3. Only the value of [tex]\( a \)[/tex] is a perfect square: Similarly, [tex]\( a \)[/tex] being a perfect square is not a requirement for the general form of the equation [tex]\( a(x^2) - c = 0 \)[/tex].
4. The value [tex]\( b = 0 \)[/tex]: This aligns perfectly with our derived structure for the difference of squares equation. As shown, [tex]\( b \)[/tex] must be zero for the equation to lack the [tex]\( bx \)[/tex] term.
Therefore, the correct answer to the question is:
The value [tex]\( b = 0 \)[/tex].
