Original Expression: \(\frac{1}{16}x^2 – \frac{1}{25}\) // Given expression
Step 1: Identify the expression as a difference of squares // Recognize the pattern
Step 2: Write as \(\frac{x^2}{4^2} – \frac{1^2}{5^2}\) // Rewrite using squares of 4 and 5
Step 3: Rewrite as \(\left(\frac{x}{4}\right)^2 – \left(\frac{1}{5}\right)^2\) // Express as squares of fractions
Step 4: Apply the difference of squares formula: \(a^2 – b^2 = (a + b)(a – b)\) // Use the formula
Step 5: Substitute \(a = \frac{x}{4}\) and \(b = \frac{1}{5}\) into the formula // Plug in values
Step 6: \(\left(\frac{x}{4} + \frac{1}{5}\right)\left(\frac{x}{4} – \frac{1}{5}\right)\) // Apply the formula
Factored Expression: \(\left(\frac{x}{4} + \frac{1}{5}\right)\left(\frac{x}{4} – \frac{1}{5}\right)\) // Final factored form
Original Expression: \(\frac{1}{9}x^2 – \frac{1}{4}\) // Given expression
Step 1: Identify the expression as a difference of squares // Recognize the pattern
Step 2: Write as \(\frac{x^2}{3^2} – \frac{1^2}{2^2}\) // Rewrite using squares of 3 and 2
Step 3: Rewrite as \(\left(\frac{x}{3}\right)^2 – \left(\frac{1}{2}\right)^2\) // Express as squares of fractions
Step 4: Apply the difference of squares formula: \(a^2 – b^2 = (a + b)(a – b)\) // Use the formula
Step 5: Substitute \(a = \frac{x}{3}\) and \(b = \frac{1}{2}\) into the formula // Plug in values
Step 6: \(\left(\frac{x}{3} + \frac{1}{2}\right)\left(\frac{x}{3} – \frac{1}{2}\right)\) // Apply the formula
Factored Expression: \(\left(\frac{x}{3} + \frac{1}{2}\right)\left(\frac{x}{3} – \frac{1}{2}\right)\) // Final factored form
