Mathematics - Quadratic Equation Question with Solution | TestHub

Let the roots be ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_8$

$\begin{array}{cc}\Rightarrow & {\alpha }_1+{\alpha }_2+\dots +{\alpha }_8=4\\ & \end{array}$

${\alpha }_1{\alpha }_2\dots .{\alpha }_8=\frac{1}{2^8}$

$\Rightarrow {\left({\alpha }_1{\alpha }_2\dots .{\alpha }_8\right)}^{1/8}=\frac{1}{2}=\frac{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_8}{8}$

$\Rightarrow \mathrm{AM}=\mathrm{GM}\Rightarrow$ all the roots are equal to $\frac{1}{2}$.

$\begin{array}{rlrl}\Rightarrow a_1& =-{}^8{C}_7{\left(\frac{1}{2}\right)}^7=-\frac{1}{2^4}{a}_2& & \end{array}$

$={}^8{C}_6{\left(\frac{1}{2}\right)}^6=-\frac{7}{2^4}{a}_3=-{}^8{C}_5{\left(\frac{1}{2}\right)}^5$