Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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