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Factoring is a technique used to break down expressions into simpler, multiplied factors. In the context of quadratic equations, factoring helps simplify the equation so that it can be solved easily. A quadratic equation typically follows the form:

• \( ax^2 + bx + c = 0 \)

Factoring involves rewriting the quadratic expression as a product of its factors. For the equation \(2x^2 - 7x = 0\), you can factor it by recognizing that both terms share a common factor, which is \(x\).
When you factor out \(x\), the equation becomes:

• \( x(2x - 7) = 0 \)

This step is crucial because it transforms the quadratic equation into a multiplication problem, setting the stage for using the Zero Product Property. Factoring requires knowing how to identify the greatest common factor and using it to simplify expressions.

The Zero Product Property is a fundamental rule in mathematics that states if the product of two numbers is zero, then at least one of the numbers must be zero. Mathematically, if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \).
In quadratic equations, once you've factored the expression, the Zero Product Property allows you to set each factor to zero separately. In our example, the factored equation \( x(2x - 7) = 0 \) uses this property.
Applying it, we form two new equations:

• \( x = 0 \)
• \( 2x - 7 = 0 \)

By solving these simple equations individually, we find potential values of \( x \) that satisfy the original quadratic equation. This property is a powerful tool when dealing with equations that have been factored into products.

Solving equations is the process of finding all values of the variable that make the equation true. After factoring the quadratic equation \( 2x^2 - 7x = 0 \) into \( x(2x - 7) = 0 \), and understanding the Zero Product Property, solving becomes straightforward:

• The first equation, \( x = 0 \), suggests that one solution is \( x = 0 \).
• For the second equation, \( 2x - 7 = 0 \), solving for \( x \) involves isolating the variable.

To solve \( 2x - 7 = 0 \):

• Add 7 to both sides to get \( 2x = 7 \).
• Divide by 2, resulting in \( x = \frac{7}{2} \).

Thus, the solutions to the original equation are \( x = 0 \) and \( x = \frac{7}{2} \). Solving equations after factoring and applying the Zero Product Property simplifies the process and ensures that all possible solutions are found efficiently.