College

Simplify [tex]\left(7x^2+3\right)\left(7x^2-3\right)[/tex] using the difference of squares formula.

A) [tex]7x^4-9[/tex]
B) [tex]49x^4-9[/tex]
C) [tex]49x^2-9[/tex]
D) [tex]7x^2-1[/tex]

Answer :

To simplify [tex]\((7x^2 + 3)(7x^2 - 3)\)[/tex] using the difference of squares formula, let's first understand the formula itself.

The difference of squares formula states that for any two numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
(a + b)(a - b) = a^2 - b^2
\][/tex]

In this problem, we need to identify what [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are in the expression [tex]\((7x^2 + 3)(7x^2 - 3)\)[/tex]:

- [tex]\(a\)[/tex] is [tex]\(7x^2\)[/tex]
- [tex]\(b\)[/tex] is [tex]\(3\)[/tex]

Now let's apply the formula:

1. Calculate [tex]\(a^2\)[/tex]. Since [tex]\(a = 7x^2\)[/tex],
[tex]\[
a^2 = (7x^2)^2 = 49x^4
\][/tex]

2. Calculate [tex]\(b^2\)[/tex]. Since [tex]\(b = 3\)[/tex],
[tex]\[
b^2 = 3^2 = 9
\][/tex]

3. Apply the difference of squares formula:
[tex]\[
a^2 - b^2 = 49x^4 - 9
\][/tex]

Therefore, the simplified expression is [tex]\[49x^4 - 9\][/tex].

The correct answer is:
B) [tex]\(49x^4 - 9\)[/tex]