In step 4, I mentioned setting each factor equal to zero. This is based on the Zero Product Property, which states that if a product of two factors is equal to zero, then at least one of the factors must be zero.

Let's break it down:

Understanding the Zero Product Property:

Given an equation in the form: $a \cdot b = 0$ The Zero Product Property tells us that: $\text{Either } a = 0 \quad \text{or} \quad b = 0$ or both.

Applying it to Your Problem:

In step 3, the quadratic expression was factored as: $x(x - 3) = 0$

Here, $x$ and $x - 3$ are the two factors. According to the Zero Product Property, for their product to be zero, either of the following must be true:

1. $x = 0$
2. $x - 3 = 0$

Solving Each Factor:

1. If $x = 0$, then we have one solution.
2. If $x - 3 = 0$, add 3 to both sides to solve for $x$: $x = 3$ This gives us the second solution.

Conclusion:

So, setting each factor equal to zero allows us to solve for the possible values of $x$, which in this case are $x = 0$ and $x = 3$.

Does this explanation help clarify the concept? Would you like to explore further examples or details? Here are some questions that might help deepen your understanding:

1. What is the significance of the Zero Product Property in solving quadratic equations?
2. Can you apply the Zero Product Property to more complex factorizations?
3. How do you check if a quadratic expression is factorable?
4. What should you do if a quadratic expression is not easily factorable?
5. How would the process change if the quadratic equation had complex solutions?

Tip: When solving quadratic equations, always remember to first factor the equation and then apply the Zero Product Property to find the roots.