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