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The process of solving quadratic equations is a fundamental skill in algebra. Quadratic equations are polynomials that follow the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. The solution to these equations often involves finding the values of \( x \) that make the equation true.

To solve quadratic equations, one might use several methods, including factoring, completing the square, using the quadratic formula, or graphing. However, factoring is often the quickest approach when applicable. When the quadratic can be factored into the product of two binomials, as in \( (y-2)(y-3)=0 \) from the exercise, you can apply the Zero-Factor Property to find the solutions. This method is efficient because it reduces the problem to simpler linear equations, allowing for a swift resolution.

Factoring polynomials is like breaking down a number into its prime factors; it involves finding an equivalent expression that is a product of simpler polynomials. In the case of quadratic equations, we specifically look for a product of linear factors. These linear factors usually represent the equation's solutions.

For instance, the quadratic equation \( y^2 - 5y + 6 = 0 \) can be factored into \( (y-2)(y-3) = 0 \) because \( -2 \) and \( -3 \) are the numbers that add up to the coefficient of \( y \) (which is \( -5 \) in this case), and multiply to the constant term \( 6 \) in the original quadratic equation. This method is effective when the coefficients are relatively simple numbers, and it easily leads to the next step of applying the Zero-Factor Property.

Algebraic properties are the rules that govern the manipulation of algebraic expressions. Understanding these properties allows you to solve equations systematically and verify solutions. One such property is the Zero-Factor Property, critical for solving quadratic equations through factoring.

The Zero-Factor Property states that if the product of two algebraic expressions equals zero, then at least one of the expressions must be zero. This principle is applied in the provided exercise, where \( (y-2)(y-3)=0 \) must mean that either \( y-2=0 \) or \( y-3=0 \) because no other number besides zero can multiply by another number to result in zero. It is essential in simplifying the solving process, as it allows us to isolate each factor to solve for the unknown variable. Understanding this and other algebraic properties is crucial for anyone looking to master algebra.