https://file.aiquickdraw.com/mathbot/file/1725897712204a7emihlx.jpg

The equation given in the image is:

\[
(x + a)^3 = x^3 + px^2 + qx + 64
\]

We need to solve for the constants \( p \) and \( q \).

### Step-by-Step Solution

1. **Expand the Left Side**:  
   Let's expand the left side of the equation using the binomial expansion formula:

   \[
   (x + a)^3 = x^3 + 3ax^2 + 3a^2x + a^3
   \]

2. **Equate Coefficients**:  
   Now, compare this expansion with the right side of the equation:

   \[
   x^3 + 3ax^2 + 3a^2x + a^3 = x^3 + px^2 + qx + 64
   \]

   Equating the coefficients of like terms on both sides:

   - Coefficient of \(x^2\): \(3a = p\)
   - Coefficient of \(x\): \(3a^2 = q\)
   - Constant term: \(a^3 = 64\)

3. **Solve for \( a \), \( p \), and \( q \):**  
   - From \(a^3 = 64\), we find:
   
     \[
     a = \sqrt[3]{64} = 4
     \]

   - Substitute \(a = 4\) into the expressions for \(p\) and \(q\):

     \[
     p = 3a = 3 \times 4 = 12
     \]

     \[
     q = 3a^2 = 3 \times 4^2 = 3 \times 16 = 48
     \]

### Solution:
- \( p = 12 \)
- \( q = 48 \)

Would you like more details on the binomial expansion or have any questions? Here are some related questions:

1. How does the binomial theorem work for higher powers?
2. What are some other ways to expand polynomials?
3. What if we had a different cubic equation to compare with?
4. How do we solve for constants in equations involving polynomials?
5. What if the constants are complex numbers?

**Tip:** Always equate coefficients of like terms when comparing polynomials to solve for unknowns efficiently.

(x + a)^3 = x^3 + px^2 + qx + 64 for all values of x. Solve for p and q.

Learn how to solve the polynomial equation (x + a)^3 = x^3 + px^2 + qx + 64 by using binomial expansion and equating coefficients. This problem explores key algebraic concepts, including the binomial theorem. Suitable for students in Grades 9-12.

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