Difference between max and min value of algebraic expression

From $2pq - 20 = 52 - 2pq$: $4pq = 72 \implies pq = 18$

Now, $p^2 + q^2 - 29 = 2(18) - 20 = 16 \implies p^2 + q^2 = 45$.

We know $(p - q)^2 = p^2 + q^2 - 2pq = 45 - 36 = 9 \implies p - q = \pm 3$.

Now, $p^3 - q^3 = (p - q)(p^2 + pq + q^2) = (p - q)(45 + 18) = 63(p - q)$.

• Maximum value of $p^3 - q^3 = 63 \times 3 = 189$
• Minimum value of $p^3 - q^3 = 63 \times (-3) = -189$

Difference $= 189 - (-189) = 378$.