Let $\alpha$ and $\beta$ be the roots of $x^2+bx+1=0$. Then, the equation whose roots are $-\left(\displaystyle \alpha +\frac{1}{\beta}\right)$ and $-\left(\displaystyle \beta +\frac{1}{\alpha}\right)$, is?
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$x^2=0$
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$x^2+2bx+4=0$
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$x^2-2bx+4=0$
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$x^2-bx+1=0$
C
Correct answer
Explanation
Roots of x^2 + bx + 1 = 0 are alpha, beta. alpha + beta = -b, alpha*beta = 1. New roots are S = -(alpha + 1/beta) and P = -(beta + 1/alpha). S = -( (alpha*beta + 1) / beta ) = -2/beta. P = -( (beta*alpha + 1) / alpha ) = -2/alpha. Sum of new roots = -2(alpha + beta) / (alpha*beta) = -2(-b)/1 = 2b. Product of new roots = 4 / (alpha*beta) = 4. Equation: x^2 - (sum)x + (product) = x^2 - 2bx + 4 = 0.