Factor completely: $x^2-25$
The given expression is \(x^2 - 25\). This is a binomial expression where both terms are squares: \(x^2\) is the square of \(x\), and \(25\) is the square of \(5\). This structure matches the algebraic identity for the difference of two squares, which states that for any real numbers \(a\) and \(b\), the difference of their squares can be expressed as:
\[a^2 - b^2 = (a - b)(a + b)\]
To factor \(x^2 - 25\), we identify \(a = x\) and \(b = 5\). By substituting these values into the difference of squares identity, we get:
\[x^2 - 25 = x^2 - 5^2 = (x - 5)(x + 5)\]
This factorization shows that the expression is equivalent to the product of \((x - 5)\) and \((x + 5)\). Comparing this to the given options, we find that it corresponds to option B: \((x + 5)(x - 5)\).
Factoring expressions like \(x^2 - 25\) is a fundamental skill in algebra with many real-world applications. For example, if an architect is designing a rectangular area where the length is increased by 5 meters and the width is decreased by 5 meters from a square base of side length \(x\), the new area is \((x + 5)(x - 5) = x^2 - 25\). This technique simplifies complex polynomial equations, making them easier to solve in fields like physics, engineering, and economics.