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When you encounter a quadratic equation, factoring is a technique you can use to simplify and solve it. Factoring involves expressing the equation in a product form. So, instead of having terms added or subtracted, we break them into a multiplication of two or more simpler expressions that, when expanded, return to the original polynomial.

Quadratics, being second-degree polynomials, often contain a setup where you recognize patterns or common terms. In the given exercise, the equation \[-2x^2 + 3x = 0\]is already arranged. You notice that each term shares a common factor, which in this case is "x". By factoring out the common element, the quadratic simplifies into \[x(-2x + 3) = 0\].

This expression is now easier to manage, having broken the original quadratic down into simpler linear components. Remember, factoring is like peeling away at layers to expose the base elements that build up the equation.

Finding the greatest common factor (GCF) is crucial in simplifying polynomials, particularly in factoring. The GCF is the largest factor shared by each term in the polynomial.

In the quadratic equation at hand, \(-2x^2 + 3x = 0\),we identify the GCF by looking at what each term has in common. Here, both terms include the variable "x", giving us "x" as the greatest common factor.

Extracting this factor simplifies the quadratic from \(-2x^2 + 3x\) to \(x(-2x + 3)\).

• This extraction makes the process of solving the equation more direct and understandable.
• It often leads to separating the equation into linear factors, which are easier to solve.

Identifying and factoring out the GCF is a strategic way to break down complex equations, ensuring you are working with the simplest possible form. It can be a critical first step in the process of solving these types of mathematical problems.

Solving equations, particularly quadratic ones, involves finding their roots or solutions. After you've factored a quadratic, like with \[x(-2x + 3) = 0\], you set each factor equal to zero. This step derives from the "Zero Product Property," which states that if a product of multiple terms equals zero, at least one of the terms must be zero.

This gives us two separate equations to solve:

• \(x = 0\)
• \(-2x + 3 = 0\)

Solving the first equation, \(x = 0\), is straightforward since it's isolated. For the second equation, rearrange and solve for "x" to find \(-2x = -3\) and thus: \(x = 1.5\).

This process highlights how breaking down complex expressions into simpler ones can simplify solving them. You not only identify all possible solutions but also improve comprehension of each part of the original equation. Approaching equations piece by piece, especially after factoring, is a robust strategy for revealing their solutions with ease.