91Ó°ÊÓ

Factoring quadratic equations involves rewriting the equation in a form that is easier to solve. Quadratic equations usually have the structure \( ax^2 + bx + c = 0 \). Here, the goal is to express the quadratic as a product of two binomials. In simpler cases, like in the equation \( x^2 + x = 0 \), you can use common factoring methods.

**What is Common Factoring?**

In the exercise example, the equation \( x^2 + x = 0 \) can be factored by identifying the common factor in each term. Notice that \( x \) is the common factor in both terms.

- First term: \( x^2 \) has \( x \) as a factor.
- Second term: \( x \) is obviously present.

By factoring out \( x \), we get \( x(x + 1) = 0 \). This step is crucial, as it prepares the equation for the next solving stage, using the Zero-Product Property.

The Zero-Product Property is a fundamental concept used in solving factored quadratic equations. This property states that if the product of two numbers is zero, then at least one of the numbers must be zero.

**Applying the Zero-Product Property:**

Once you have your equation in factored form, like \( x(x + 1) = 0 \), you can apply this property.

- For the product to equal zero, set each factor to zero. You break it into two separate equations: \( x = 0 \) and \( x + 1 = 0 \).

- Solve these simpler equations independently.

This step reduces a possibly complex quadratic equation into more manageable linear equations, making it simpler to find the solutions.

Once the quadratic equation is factored and the Zero-Product Property is applied, the solutions are found. These are the values of \( x \) that satisfy the original equation after going through the previous steps.

**Solution Process:**

From the equations \( x = 0 \) and \( x + 1 = 0 \), solve each one to get the solutions:

• For \( x = 0 \), it's straightforward—this is already a solution.
• For \( x + 1 = 0 \), subtract 1 from both sides to find \( x = -1 \).

The final solutions of the quadratic equation \( x^2 + x = 0 \) are \( x = 0 \) and \( x = -1 \). These are the values that satisfy the original equation, representing where the function crosses or touches the x-axis on a graph.