Exercise 3.9 1. Factorize the following. a) \( 49 x^{2}-9 y^{2} \)

To factor \( 49x^{2} - 9y^{2} \), we can recognize it as a difference of squares. This can be expressed as \( (7x)^{2} - (3y)^{2} \). Using the difference of squares formula, \( a^{2} - b^{2} = (a-b)(a+b) \), we rewrite it as: \[ (7x - 3y)(7x + 3y) \] So, the expression factors to \( (7x - 3y)(7x + 3y) \). Next, if we want to look at how this applies in real life, consider how factoring helps in simplifying algebraic expressions in fields like physics or engineering, where equations often need to be solved quickly and efficiently. Simplifying equations can lead to faster calculations and better insights into the relationships between variables. Additionally, when working on problems like this one, a common mistake is to overlook applying the difference of squares concept. Always look for patterns and see if the expression can be rewritten in the form of \( a^{2} - b^{2} \); it’s a handy shortcut that can save you time and effort!