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When we talk about solving quadratic equations by factoring, we are referring to a process in which a quadratic equation of the form ax^2 + bx + c = 0 is expressed as a product of two binomials. Factoring involves finding two numbers that multiply to give ac (the product of the coefficient of x^2 and the constant term) and add up to b (the coefficient of x).

For example, to factor the quadratic equation x^2 - 13x + 36 = 0, we need to find two numbers that multiply to 36 (ac) and add up to -13 (b). These numbers are -4 and -9. Hence, the factored form of the equation is (x - 4)(x - 9) = 0.

Understanding how to factor quadratics is crucial as it simplifies finding the equation's roots. Moreover, mastering this technique can help solve more complicated algebraic expressions and is foundational for higher-level math studies.

The zero product property is an essential principle in algebra that becomes particularly useful after factoring a quadratic equation. The property states that if the product of two factors is zero, then at least one of the factors must be zero. Mathematically, if ab = 0, then either a = 0, b = 0, or both.

In the context of the provided quadratic equation, after factoring, we have (x - 4)(x - 9) = 0. According to the zero product property, we set each factor equal to zero (x - 4 = 0 and x - 9 = 0) to find the values of x that satisfy the equation. This property provides a straightforward method to determine all possible solutions once a quadratic has been factored.

The solutions or roots of a quadratic equation are the values of x that satisfy the equation ax^2 + bx + c = 0. When we factor the equation and apply the zero product property, as in the previous sections, the solutions are the values that make each binomial factor equal to zero. In fact, the number of times a particular solution occurs is called its multiplicity, which in basic quadratic equations like ours will be one.

In our example, we have determined through factoring that the solutions to the equation x^2 - 13x + 36 = 0 are x = 4 and x = 9. These solutions are the points where the graph of the quadratic function intersects the x-axis. Therefore, understanding quadratic solutions is not just about solving an equation; it's about understanding the behavior of quadratic functions visually and algebraically.