What is the completely factored form of $2 x^2-32$?
We are given the expression \(2x^2 - 32\) and we need to find its completely factored form. The first step in factoring algebraic expressions is to check if there is a common factor among all terms. Here, we can see that both 2 and 32 are divisible by 2. Thus, we can factor out 2 from the expression:
\[2x^2 - 32 = 2(x^2 - 16).\]
Now, we look at the expression inside the parentheses: \(x^2 - 16\). We recognize this as a difference of squares, which follows the algebraic identity:
\[a^2 - b^2 = (a - b)(a + b).\]
In our case, \(x^2 - 16\) can be written as \(x^2 - 4^2\), where \(a = x\) and \(b = 4\). Applying the identity, we get:
\[x^2 - 16 = (x - 4)(x + 4).\]
Finally, we combine the common factor we pulled out in the first step with the result of factoring the difference of squares:
\[2x^2 - 32 = 2(x - 4)(x + 4).\]
Comparing this result with the given options, we see that it matches option c. Therefore, the completely factored form of \(2x^2 - 32\) is \(2(x+4)(x-4)\).
Factoring algebraic expressions is a fundamental skill used in various real-life scenarios, such as simplifying physical formulas or optimizing design parameters. For instance, if you are designing a rectangular garden and the area is given by the expression \(2x^2 - 32\), factoring it into \(2(x-4)(x+4)\) could help you determine the potential length and width of the garden based on the variable \(x\), ensuring that the dimensions remain positive and realistic. Understanding these patterns allows engineers, architects, and data analysts to break down complex systems into simpler, more manageable components.