Question

if alpha and beta are the roots of the quadratic equation x2-7x+10=0 then the quadratic equation whose roots are 1/alpha and 1/beta is

Answer 1

Dear student
$$\begin{gathered} {\text{Since α and β are the zeros of x}}^2-7\mathrm{x}+10=0 \\ \mathrm{So},\mathrm{\alpha }+\mathrm{\beta }=\frac{-\mathrm{coeff}.\mathrm{of}\mathrm{x}}{\mathrm{coeff}.\mathrm{of}{\mathrm{x}}^2}=7 \\ \mathrm{and}\mathrm{\alpha \beta }=\frac{\mathrm{Constant}\mathrm{term}}{\mathrm{coeff}.\mathrm{of}{\mathrm{x}}^2}=10 \\ \mathrm{Let}\mathrm{S}\mathrm{and}\mathrm{P}\mathrm{be}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{zeros}\mathrm{and}\mathrm{product}\mathrm{of}\mathrm{the}\mathrm{zeros}\mathrm{respectively}\mathrm{of}\mathrm{the} \\ \mathrm{required}\mathrm{quadratic}\mathrm{equation} \\ \mathrm{S}=\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}=\frac{\mathrm{\alpha }+\mathrm{\beta }}{\mathrm{\alpha \beta }}=\frac{7}{10} \\ \mathrm{P}=\frac{1}{\mathrm{\alpha }}\times \frac{1}{\mathrm{\beta }}=\frac{1}{\mathrm{\alpha \beta }}=\frac{1}{10} \\ \mathrm{So},\mathrm{required}\mathrm{polynomial}\mathrm{is}\mathrm{k}\left({\mathrm{x}}^2-\mathrm{Sx}+\mathrm{P}\right)=\mathrm{k}\left({\mathrm{x}}^2-\frac{7}{10}\mathrm{x}+\frac{1}{10}\right)=\mathrm{k}\left(10{\mathrm{x}}^2-7\mathrm{x}+1\right)\mathrm{where}\mathrm{k}\mathrm{is}\mathrm{any}\mathrm{non}-\mathrm{zero}\mathrm{constant}. \end{gathered}$$
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