If p³ - q³ = (p - q) (p - q)² - xpq, then find the value of x :

Solution(By Examveda Team)

$${p^3} - {q^3} = \left( {p - q} \right)$$   $$\left\{ {{{\left( {p - q} \right)}^2} - xpq} \right\}$$
$$ \Rightarrow \left( {p - q} \right)\left( {{p^2} + {q^2} + pq} \right)$$     $$ = \left( {p - q} \right)$$   $$\left\{ {{{\left( {p - q} \right)}^2} - xpq} \right\}$$
$$\left\{ {\because {a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)} \right\}$$
By cancelling same terms of both sides
$$ \Rightarrow {p^2} + {q^2} + pq = {p^2} + {q^2}$$     $$ - 2pq - xpq$$   $$\left\{ {{{\left( {a - b} \right)}^2} = {a^2} + {b^2} - 2ab} \right\}$$
$$\eqalign{ & \Rightarrow 3pq = - xpq \cr & \Rightarrow x = - 3 \cr} $$