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When you come across a quadratic equation like the one in our exercise, \(2x^{2}+10x=0\), the first step you typically take is to factor the equation. Factoring quadratics involves expressing the equation as a product of its factors. To simplify things, look for common terms in each part of the quadratic. In this case, both terms have an \(x\), which simplifies our task.

Once you notice that \(x\) is a common factor, you can factor it out, giving you \(x(2x+10)=0\). Factoring is a crucial skill because it reduces complex equations to a simpler form where the zero product property can then be applied, paving the way to finding the roots of the quadratic equation.

Factoring isn't always straightforward and can sometimes require more techniques such as splitting the middle term or using the quadratic formula, but with practice, it becomes an essential tool in your algebra toolkit.

The zero product property is essentially our golden ticket to solving quadratic equations once they've been factored into a product form. It states that if a product of two factors is zero, one or both of the factors must be zero. This property is very straightforward but extremely powerful.

In the equation \(x(2x+10)=0\), you have two factors: \(x\) and \(2x + 10\). According to the zero product property, for this equation to hold true, either \(x\) equals zero, or \(2x + 10\) equals zero. This is why in Step 2 of the provided solution, the equation is split into \(x = 0\) and \(2x + 10 = 0\).

By applying this logic, we can find the values of \(x\) that satisfy the condition of the product being zero. It's a process that can and often should be visualized step-by-step, ensuring a clear understanding of how we arrive at the final solution.

Algebra can sometimes feel like a puzzle, providing an opportunity for a step-by-step approach to solving problems. This structured, methodical process is beneficial because it gives a clear path to follow and ensures that no steps are missed, improving comprehension and retention of the concepts involved.

Understanding Each Step

For our problem \(2x^{2}+10x=0\), the first step is factoring, which simplifies the equation. The subsequent step involves using the zero product property to set up equations that are simpler and can be solved directly. Finally, we isolate \(x\) to find the solutions.

Why It Works

Following a step-by-step approach not only helps in systematically reaching the solution, but also in identifying potential errors along the way. This strategy applies whether you're just learning the ropes or tackling more complex algebraic equations. The provided solution shows the practical application of this approach and underscores its effectiveness in learning and teaching algebra.