`(2-50x ^(2))=` ?

Options

A. `2 (1 - 5x )^(2)`
B. `2 (1 + 5x)^(2)`
C. `(2-5x) (2+ 5x ) `
D. `2(1 -5 x) (1 + 5x)`

Answer

To factor the expression \(2 - 50x^2\), we can follow these steps:

### Step 1: Identify the common factor
The first step is to identify if there is a common factor in the terms of the expression \(2\) and \(50x^2\). Here, we can factor out \(2\).

\[ 2 - 50x^2 = 2(1 - 25x^2) \]

### Step 2: Recognize the difference of squares
Next, we notice that \(1 - 25x^2\) can be recognized as a difference of squares. The expression can be rewritten as:

\[ 1 - (5x)^2 \]

### Step 3: Apply the difference of squares formula
We can apply the difference of squares formula, which states that \(a^2 - b^2 = (a + b)(a - b)\). Here, \(a = 1\) and \(b = 5x\).

Using the formula, we get:

\[ 1 - (5x)^2 = (1 + 5x)(1 - 5x) \]

### Step 4: Combine the factors
Now, we substitute back into our expression:

\[ 2(1 - 25x^2) = 2(1 + 5x)(1 - 5x) \]

### Final Answer
Thus, the fully factored form of the expression \(2 - 50x^2\) is:

\[ 2(1 + 5x)(1 - 5x) \]

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