Question

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

Answer 1

Let the quantity to be added be x.
Then
`((a + b) + x)/((a - b) + x) = (a + b)^2/(a - b)^2`
⇒ (a + b) (a - b)² + (a - b)².x
= (a + b)² (a - b) + (a + b)².x
⇒ [(a + b)² - (a - b)²]x
= (a² - b²) (a - b) - (a² - b²) (a + b)
⇒ (4abx) = (a² - b²) [(a - b) - (a + b)]
⇒ x = `(-2b(a^2 - b^2))/(4ab) = (b^2 - a^2)/(2a)`.