Question

If $\alpha$ and $\beta$ are the roots of the equation $x^2+px+q=0$, then $-{\alpha }^{-1},-{\beta }^{-1}$ are the roots of which one of the following equations?

• A. $q{x}^2-px+1=0$
• B. $q{x}^2+px+1=0$
• C. $x^2+px-q=0$
• D. $x^2-px+q=0$

Answer 1

Answer: (A)

$q{x}^2-px+1=0$

$\alpha ,\beta$ are the roots of $x^2+px+q=0....\left(i\right)$
To obtain the equation whose roots are $-\frac{1}{\alpha },-\frac{1}{\beta }$, we change $x$ into $-\frac{1}{x}$ in (i).
Therefore, required equation is ${(-\frac{1}{x})}^2+p.(-\frac{1}{x})+q=0$.
Thus the equation is $q{x}^2-px+1=0$.