How to know what number to pick for factoring 

When factoring a quadratic expression like \(63n^2 + 14n = 0\), the first step often involves finding the **Greatest Common Factor (GCF)** of all the terms. The GCF is the largest factor that divides each term in the expression evenly.

### Steps to Identify the Number for Factoring

1. **List the Factors of Each Coefficient:**
   - For \(63n^2\), the coefficient is 63. The factors of 63 are \(1, 3, 7, 9, 21, 63\).
   - For \(14n\), the coefficient is 14. The factors of 14 are \(1, 2, 7, 14\).

2. **Identify the Common Factors:**
   - Look for common factors in the lists above. The common factors between 63 and 14 are \(1\) and \(7\).

3. **Choose the Greatest Common Factor (GCF):**
   - The largest common factor is \(7\). This is the number you'll use for factoring out.

4. **Include Any Common Variables:**
   - Both terms in the equation have the variable \(n\). So, you also factor out the smallest power of \(n\) present in all terms, which is \(n\).

### Example in Context

For the equation \(63n^2 + 14n\):
- **GCF of the coefficients**: 7
- **Common variable**: \(n\)

Thus, the GCF of the entire expression is \(7n\), and you factor that out:

\[ 63n^2 + 14n = 7n(9n + 2) \]

### Why This Works
Factoring simplifies the equation into a product of simpler expressions, making it easier to solve. In this case, factoring out \(7n\) transforms the quadratic into a linear equation and a simpler quadratic.

### Additional Tips for Factoring

1. **Check Each Term:** Always consider all terms in the expression when determining the GCF.
2. **Simplify First:** If possible, simplify the equation before factoring, such as by dividing through by a common factor.
3. **Prime Factorization:** For more complex coefficients, use prime factorization to determine the GCF.

Would you like more examples or have further questions?

Here are some related questions to explore:
1. What if there are no common factors—how would you factor the quadratic then?
2. How can prime factorization help in finding the GCF?
3. How do you factor a quadratic expression that doesn’t have a GCF?
4. What is the difference between factoring by grouping and factoring by finding the GCF?
5. How would you factor a quadratic trinomial, like \(ax^2 + bx + c\)?

**Tip:** Always write down the factors of the coefficients when factoring to ensure you don't miss the GCF. This makes the process smoother and more accurate.

Learn how to identify the correct number for factoring in quadratic equations with a step-by-step guide. This problem covers key algebraic concepts, including factoring and identifying the greatest common factor (GCF). Suitable for students in Grades 7-9.

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