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The Zero Product Property is a fundamental aspect of algebra that comes into play when you need to solve factored equations. Consider a scenario where you have an equation such as \( a \times b = 0 \).According to this property, either \( a = 0 \) or \( b = 0 \), or both.
By applying this property, you can simplify the process of finding solutions for the equation. For instance, when you have the equation \( 2x(x+9) = 0 \), you can determine the solutions by setting each factor to zero separately:

• \( 2x = 0 \)
• \( x + 9 = 0 \)

When you solve these, you get:

• For \( 2x = 0 \), divide both sides by 2 to find \( x = 0 \).
• For \( x + 9 = 0 \), subtract 9 from both sides to find \( x = -9 \).

Understanding the Zero Product Property helps you quickly identify solutions to quadratic equations once they are in factored form.

Finding a common factor is a powerful technique used to simplify expressions and solve equations. In the expression \( 2x^2 + 18x \),both terms have a common factor of \( 2x \). By factoring out this common factor, you can greatly simplify the equation. Here's how you do it step-by-step:

• Identify the greatest common factor (GCF), which in this case is \( 2x \).
• Write each term as a product of the GCF and another factor. For example, \( 2x^2 \) can be written as \( 2x \times x \), and \( 18x \) as \( 2x \times 9 \).
• Factor the GCF out of both terms: \( 2x(x + 9) \).

By factoring out the common factor, you simplify the equation and make it ready for solving using the Zero Product Property. This technique not only makes equations more manageable but also reveals potential solutions more easily.

A quadratic equation is a polynomial equation of the second degree, which generally takes the form\( ax^2 + bx + c = 0 \).In our example, the equation is already set to zero: \( 2x^2 + 18x = 0 \).The fundamental characteristic of a quadratic equation is that it can be expressed as the product of two binomials when it is factorable.
Once a quadratic equation is factored, solving it involves finding the values of \( x \) that make the equation true. These values are called the roots or solutions of the equation. You typically use methods like factoring, the quadratic formula, or completing the square to find these solutions.
In the given equation, after factoring the common term \( 2x \), it becomes\( 2x(x + 9) = 0 \).Each factor, \( 2x \) and \( x+9 \), represents potential solutions, which can be found by setting each equal to zero, demonstrating the power and simplicity of working with quadratic equations in their factored form.