91Ó°ÊÓ

Factoring is an essential tool when solving quadratic equations, especially when you want to simplify the expression and make solving easier. In the given problem, the equation is \(4x^2 + 8x = 0\). To find a common factor, look at each term separately. Both \(4x^2\) and \(8x\) have a common factor of \(4x\). You'll want to factor this out as it simplifies the equation by reducing it to a product form.

Here's how you do it:

• Identify common factors in the terms. Both terms share \(4x\).
• Factor \(4x\) out of each term, changing the expression to \(4x(x+2)=0\).

Factoring sets the stage for the application of further algebraic techniques by simplifying the complexity of an equation. It helps in breaking down complex algebraic expressions into more manageable parts.

The Zero Product Property is a fundamental principle when working with equations that have products, like the one we have after factoring: \(4x(x+2) = 0\). This principle states that if a product of two factors is zero, then at least one of the factors must be zero. This is a critical step in solving algebraic equations.

In our problem, once we have the equation in factored form, \(4x(x+2) = 0\), we can apply this property by breaking the equation into two separate equations:

• Set \(4x = 0\)
• Set \(x + 2 = 0\)

This approach allows you to solve for the variable \(x\) by focusing on each factor individually, reducing complexity and clarifying the problem.

Solving equations is the final step, translating our work from understanding to finding actual solutions. After applying the Zero Product Property, we have two simpler equations that we can solve separately. Let's take each one.

1. **Solve \(4x = 0\):** - Divide both sides by \(4\) to isolate \(x\). - This leads to \(x = 0\).2. **Solve \(x + 2 = 0\):** - Subtract \(2\) from both sides to isolate \(x\). - This results in \(x = -2\).

By solving each equation, you find all potential solutions for the original problem. Thus, the solutions to the equation \(4x^2 + 8x = 0\) are \(x = 0\) and \(x = -2\). These steps ensure all possible values for \(x\) that satisfy the original equation are accounted for.