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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 ?
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Solution
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.
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