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If alpha and beta are the roots of x^2 -...

If `alpha` and `beta` are the roots of `x^2 - p (x+1) - c = 0`, then the value of `(alpha^2 + 2alpha+1)/(alpha^2 +2 alpha + c) + (beta^2 + 2beta + 1)/(beta^2 + 2beta + c)`

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The given equation os `x^(2) - px - (p + c) = 0 ` . Therefore,
` alpha + beta = p, alpha beta = - (p + c)`
So, ` (alpha + 1 ) (beta + 1) = alpha beta + (alpha + beta) + 1`
` - (p + c) + p + 1`
` = 1 - c ` (1)
Now , `(alpha^(2) + 2 alpha + 1 )/(alpha^(2) + 2alpha + c) + (beta^(2) +2 beta + 1)/(beta^(2) + 2 beta + c)`
` = ((alpha + 1)^(2))/((alpha + 1)^(2) - (1 - c)) + ((beta + 1)^(2))/((beta + 1)^(2) - (1 - c)) `
` = ((alpha + 1)^(2))/((alpha + 1)^(2) - (alpha + 1)(beta +1)) + ((beta + 1)^(2))/((beta + 1)^(2) - (alpha + 1 )(beta +1)) ` [Using (1)]
`(alpha + 1)/(alpha - beta ) + (beta + 1)/(beta - alpha ) = ((alpha + 1) - (beta +1))/(alpha - beta ) = 1` .
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