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Let alpha, beta be the roots of x^(2) +...

Let `alpha, beta ` be the roots of `x^(2) + bx + 1 = 0` . Them find the equation whose roots are `-(alpha + 1//beta) and -(beta + 1//alpha).`

Text Solution

Verified by Experts

The correct Answer is:
`x^(2) - x (2b) + 4 = 0`
Since ` alpha , beta ` roots of ` x^(2) + bx + 1 = 0` , we have ` alpha + beta = -b , alpha beta = 1`
Now ,` (-alpha - (1)/(beta)) + (- beta - (1)/(alpha )) = - (alpha + beta) - ((1)/(beta + (1)/(alpha))`
= ` b + b 2b`
and `( - alpha - (1)/(beta) ( - beta - (1)/(alpha )) = alpha beta + 2 + (1)/(alpha beta ) = 1 + 2 + 1 = 4`
Thus, the equation whose roots are
` - alpha - 1//beta and - beta - 1//alpha is x^(2) - x (2b) + 4 = 0` .
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