To determine the correct equation for finding [tex]\( x \)[/tex], the smaller negative integer, given that two negative integers are 8 units apart on the number line and have a product of 308, we can follow these steps:
1. Define the Variables:
- Let [tex]\( x \)[/tex] be the smaller negative integer.
- The larger negative integer will be [tex]\( x + 8 \)[/tex], since they are 8 units apart.
2. Formulate the Relationship:
- The problem states that the product of these two integers is 308. So we can write the equation:
[tex]\[
x \cdot (x + 8) = 308
\][/tex]
3. Expand and Simplify:
- Distribute [tex]\( x \)[/tex] on the left side of the equation:
[tex]\[
x^2 + 8x = 308
\][/tex]
4. Set Up a Standard Quadratic Equation:
- To form a standard quadratic equation, we need to bring all terms to one side of the equation:
[tex]\[
x^2 + 8x - 308 = 0
\][/tex]
5. Verify the Quadratic Equation:
- The correct form of the quadratic equation based on the given conditions is:
[tex]\[
x^2 + 8x - 308 = 0
\][/tex]
Based on this detailed reasoning, the equation that can be used to determine [tex]\( x \)[/tex], the smaller negative integer, is:
[tex]\[
\boxed{x^2 + 8x - 308 = 0}
\][/tex]
