DIFFERENCE BETWEEN SQUARES \( \begin{array}{l}x^{2}-1 \\ \frac{a^{4}}{16}-\frac{1}{25} b^{2} \\ x^{2 a}-1\end{array} \begin{array}{l}x^{2}\left(a^{2}-9\right)+y^{2}\left(-a^{2}+9\right)\end{array} \)

The **difference of squares** is a fundamental algebraic identity that allows us to factor expressions of the form \( a^2 - b^2 \) into \( (a + b)(a - b) \). This technique is widely used to simplify and solve various algebraic expressions. Let's apply this method to each of the given expressions. --- ### 1. \( x^{2} - 1 \) **Step 1:** Recognize the expression as a difference of squares. \[ x^{2} - 1 = x^{2} - 1^{2} \] **Step 2:** Apply the difference of squares formula. \[ x^{2} - 1 = (x + 1)(x - 1) \] --- ### 2. \( \frac{a^{4}}{16} - \frac{1}{25} b^{2} \) **Step 1:** Rewrite each term as a square. \[ \frac{a^{4}}{16} = \left( \frac{a^{2}}{4} \right)^{2}, \quad \frac{1}{25} b^{2} = \left( \frac{b}{5} \right)^{2} \] **Step 2:** Express the original expression as a difference of squares. \[ \frac{a^{4}}{16} - \frac{1}{25} b^{2} = \left( \frac{a^{2}}{4} \right)^{2} - \left( \frac{b}{5} \right)^{2} \] **Step 3:** Apply the difference of squares formula. \[ \frac{a^{4}}{16} - \frac{1}{25} b^{2} = \left( \frac{a^{2}}{4} + \frac{b}{5} \right) \left( \frac{a^{2}}{4} - \frac{b}{5} \right) \] **Optional:** To simplify further, you can find a common denominator for each factor. \[ \left( \frac{5a^{2} + 4b}{20} \right)\left( \frac{5a^{2} - 4b}{20} \right) = \frac{(5a^{2} + 4b)(5a^{2} - 4b)}{400} \] --- ### 3. \( x^{2a} - 1 \) **Step 1:** Recognize the expression as a difference of squares by identifying \( x^{a} \) as the base. \[ x^{2a} - 1 = \left( x^{a} \right)^{2} - 1^{2} \] **Step 2:** Apply the difference of squares formula. \[ x^{2a} - 1 = \left( x^{a} + 1 \right) \left( x^{a} - 1 \right) \] --- ### 4. \( x^{2}(a^{2} - 9) + y^{2}(-a^{2} + 9) \) **Step 1:** Simplify the expression by factoring out common terms. \[ x^{2}(a^{2} - 9) + y^{2}(-a^{2} + 9) = x^{2}(a^{2} - 9) - y^{2}(a^{2} - 9) = (a^{2} - 9)(x^{2} - y^{2}) \] **Step 2:** Notice that both \( a^{2} - 9 \) and \( x^{2} - y^{2} \) are differences of squares. \[ a^{2} - 9 = a^{2} - 3^{2} = (a + 3)(a - 3) \] \[ x^{2} - y^{2} = (x + y)(x - y) \] **Step 3:** Combine the factorizations. \[ x^{2}(a^{2} - 9) + y^{2}(-a^{2} + 9) = (a + 3)(a - 3)(x + y)(x - y) \] --- By systematically applying the difference of squares formula, we've successfully factored each of the given expressions into simpler, multiplicative forms.