If $$p^2 + q^2 - 29 = 2pq - 20 = 52 - 2pq$$, then the difference between the maximum and minimum possible values of $$(p^3 - q^3)$$ is

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Accepted Answer

1. **Simplify the Given Equations:**  
   From $$p^2 + q^2 - 29 = 2pq - 20$$ and $$2pq - 20 = 52 - 2pq$$:  
   $$
   4pq = 72 \implies pq = 18.
   $$

2. **Relation Between $$p$$ and $$q$$:**  
   Substitute into the first equation:  
   $$
   p^2 + q^2 - 29 = 2pq - 20 \implies p^2 + q^2 = 29 - 20 +36 = 45.
   $$  
   Using the identity $$p^2 + q^2 = (p-q)^2 + 2pq$$:  
   $$
   (p-q)^2 + 2(18) = 45 \implies (p-q)^2 = 45 - 36 = 9.
   $$  
   Thus, $$p-q = \pm3$$.

3. **Expression for $$(p^3 - q^3):$$**  
   Using the identity:  
   $$
   p^3 - q^3 = (p-q)(p^2 + pq + q^2).
   $$  
   Substitute values:  
   $$
   p^3 - q^3 = (p-q)(63).
   $$

4. **Maximum and Minimum Values of $$(p^3 - q^3):$$**  
   - When $$p-q = +3$$:  
     $$
     p^3 - q^3 = (3)(49) = 189.
     $$  
   - When $$p-q = -3$$:  
     $$
     p^3 - q^3 = (-3)(49) = -189.
     $$

5. **Difference Between Maximum and Minimum Values:**  
   $$
   	ext{Difference} = |189 - (-189)| = |189 + 189| = 378.
   $$

Thus, the difference between the maximum and minimum possible values of $$(p^3 - q^3)$$ is **378**.

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