Question

If x + y = `7/2  "and xy" =5/2`; find:  x - y and x² - y²

Answer 1

We know that,

(x + y)² = x² + 2xy + y²

and
(x - y)² = x² - 2xy + y²

Rewrite the above equation, we have

(x - y)² = x² + y² + 2xy - 4xy 

= (x + y)² - 4xy              ...(1)

Given that `"x + y" = 7/2  "and xy" =5/2` 

Substitute the values of (x + y) and (xy)

in equation (1), we have

(x - y)² =` (7/2)^2 - 4(5/2)`

= `49/4 - 10`

= `9/4`

⇒ x - y = `+- sqrt(9/4)`

⇒ a - b = `+-(3/2)`            ...(2)

We know that,

x² - y² = (x + y)(x - y)             ...(3)

From equation (2) we have,

x - y = `+- 3/2`

Thus, equation (3) becomes,

x² - y² = `(7/2)xx( +- 3/2)`           ...[Given x + y = `7/2`]

⇒ x² - y² = `+- 21/4`