Question

What quantity must be added to each term of the ratio (p + q) : (p - q) to make it equal to (p + q)² : (p - q)² ?

Answer 1

Let x be added from each term such that 

`(("p + q") + "x")/(("p - q") + "x") = ("p + q")^2/("p - q")^2`

⇒ (p + q + x)(p² - 2pq+ q² ) = (p² + 2pq + q²)(p- q + x) 

⇒ p³ + p² q + p²x + q²p + q³ + q²x - 2p²q - 2pq² - 2pqx 

⇒ p³ - p²q + p² x + q²p - q³ + q²x + 2p²q - 2pq² + 2pqx

⇒ p²q + q³ - 2 p²q- 2pqx = -p²q - q³ + 2p² q + 2pqx 

⇒ 2q³ - p²q = 4pqx 

⇒ 2q( q² - p² ) = 4pqx 

⇒  x = `("q"^2 - "p"^2)/(2"p")`

`("q"^2 - "p"^2)/(2"p")` should be added to each term.