To factor the quadratic equation \( p^2 - 5p + 6 = 0 \), you need to find two numbers that multiply to give the constant term (6) and add to give the coefficient of the linear term (-5).

Let's break this down:

1. **Identify the coefficients:**
   - The coefficient of \( p^2 \) (the quadratic term) is 1.
   - The coefficient of \( p \) (the linear term) is -5.
   - The constant term is 6.

2. **Find two numbers that multiply to 6 and add to -5:**
   - The numbers are -2 and -3 because:
     - \(-2 \times -3 = 6\) (product)
     - \(-2 + -3 = -5\) (sum)

3. **Write the quadratic expression as a product of two binomials:**
   \[
   p^2 - 5p + 6 = (p - 2)(p - 3)
   \]

So, the equation can be factored as:

\[
(p - 2)(p - 3) = 0
\]

4. **Solve for \( p \):**
   - Set each factor equal to zero:
     - \( p - 2 = 0 \) gives \( p = 2 \)
     - \( p - 3 = 0 \) gives \( p = 3 \)

Thus, the solutions are \( p = 2 \) and \( p = 3 \).

Would you like any further details or have any questions?

Here are some related questions to explore:
1. How do you factor \( p^2 + 5p + 6 = 0 \)?
2. What is the quadratic formula and how is it derived?
3. How can you use factoring to solve \( p^2 - 9 = 0 \)?
4. What are the steps to complete the square for \( p^2 - 5p + 6 = 0 \)?
5. How would you factor a quadratic equation with a leading coefficient greater than 1?

**Tip:** Always check your factored form by expanding it back out to ensure it matches the original quadratic expression!

Learn how to factor the quadratic equation p^2 - 5p + 6 = 0 with a detailed step-by-step solution. Understand the concept of factoring quadratic equations and solve for the roots. Suitable for Grades 7-9.

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