Factor. \( x^{2}-15 x y+26 y^{2} \)

To factor the expression \( x^{2}-15 x y+26 y^{2} \), we need to find two numbers that multiply to \( 26y^2 \) (the product of the first and last coefficients) and add up to \(-15y\) (the coefficient of the middle term). After checking, we find that the numbers \(-13y\) and \(-2y\) fit the bill! Thus, we can group the terms and factor by grouping: \[ x^{2} - 13xy - 2xy + 26y^{2} = (x - 13y)(x - 2y). \] So, the factorization is \( (x - 13y)(x - 2y) \). Additionally, if you're looking to further your understanding of factoring quadratics, exploring the concept of completing the square can be beneficial. It provides a different perspective and could be particularly helpful when the expressions do not factor neatly. This technique allows you to convert the quadratic into a perfect square trinomial, which can then be solved or analyzed in various contexts, including graphing. For practical applications, understanding how to factor quadratics can be particularly useful in solving real-world problems involving area computation. For instance, if a rectangular garden has a length represented as \( x \) and a width expressed in terms of \( y \), using factoring can help determine the dimensions that meet specific area requirements based on given conditions. It’s a handy skill for anyone interested in applying math to everyday scenarios!