To factor the expression [tex]\(4x^2 - 25\)[/tex], you can recognize it as a difference of squares. The difference of squares formula is given by [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
Step 1: Identify the squares in the expression:
- [tex]\(4x^2\)[/tex] is the square of [tex]\(2x\)[/tex] because [tex]\((2x)^2 = 4x^2\)[/tex].
- [tex]\(25\)[/tex] is the square of [tex]\(5\)[/tex] because [tex]\(5^2 = 25\)[/tex].
Step 2: Rewrite the expression using these squares:
[tex]\[4x^2 - 25 = (2x)^2 - 5^2\][/tex]
Step 3: Apply the difference of squares formula [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex] using [tex]\(a = 2x\)[/tex] and [tex]\(b = 5\)[/tex]:
[tex]\[(2x)^2 - 5^2 = (2x - 5)(2x + 5)\][/tex]
Thus, the factored form of [tex]\(4x^2 - 25\)[/tex] is:
[tex]\[
(2x - 5)(2x + 5)
\][/tex]
