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To solve the given equation \( x^2 - 36 = (a)(bx + c)(bx + d) \) and identify the numbers \( c \) and \( d \), we can follow these steps:

1. **Factorize the left-hand side**:
   The given equation is \( x^2 - 36 \). This can be factored as a difference of squares:
   \[
   x^2 - 36 = (x - 6)(x + 6)
   \]

2. **Compare with the given factored form**:
   We need to express this factorization in the form \( (a)(bx + c)(bx + d) \).

   Given that \( a \), \( b \), \( c \), and \( d \) are constants, let's assume \( a = 1 \) and \( b = 1 \) (since the quadratic term \( x^2 \) has a coefficient of 1).

   Therefore, we have:
   \[
   x^2 - 36 = (1)(1x + 6)(1x - 6)
   \]
   This simplifies to:
   \[
   (x + 6)(x - 6)
   \]

3. **Identify \( c \) and \( d \)**:
   By comparing:
   \[
   (x + 6)(x - 6) \quad \text{with} \quad (bx + c)(bx + d)
   \]
   we see that \( c = 6 \) and \( d = -6 \).

4. **Determine the bigger and smaller number**:
   Since \( c = 6 \) and \( d = -6 \):
   - The bigger number \( c \) is \( 6 \).
   - The smaller number \( d \) is \( -6 \).

So, the answers are:
- \( c = 6 \) (the bigger number)
- \( d = -6 \) (the smaller number)

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How do you factorize a difference of squares in general?
2. What is the significance of the coefficients in the factored form of a quadratic equation?
3. Can this method be applied to any quadratic equation? Why or why not?
4. What are some common mistakes to avoid when factorizing quadratic equations?
5. How can you verify that the factored form of a quadratic equation is correct?

**Tip:** Always check your factored form by expanding it back to the original equation to ensure accuracy.

x^2 - 36 = (a)(bx + c)(bx + d) is the bigger number c and is the smaller number d.

Understand how to factor the quadratic equation x^2 - 36 using the difference of squares and identify the constants c and d in the equation. This problem is designed for students in Grades 8-10 and covers essential algebraic concepts.

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