Solve each equation in Exercises 73–81 by the method of your choice. (x-3)^2 - 25 = 0

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. These equations can be solved using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the structure of quadratic equations is essential for solving them effectively.

Factoring

Factoring involves expressing a polynomial as a product of its factors. For quadratic equations, this often means rewriting the equation in a form that allows for easy identification of roots. In the given equation, (x-3)^2 - 25 can be recognized as a difference of squares, which can be factored into (x-3-5)(x-3+5) = 0, simplifying the solving process.

Difference of Squares

The difference of squares is a specific algebraic identity that states a^2 - b^2 = (a - b)(a + b). This concept is crucial when dealing with equations that can be expressed in this form, as it allows for straightforward factoring. In the context of the given equation, recognizing (x-3)^2 - 25 as a difference of squares facilitates finding the solutions quickly.