Consider the quadratic equation $\left({\log }_{10}8\right)x^2-\left({\log }_{10}5\right)x=2{\left({\log }_{2}10\right)}^{-1}-x$.  Which of the following quantities are irrational

$\left({\log }_{10}\right)x^2-\left({\log }_{10}5\right)x=2{\left({\log }_{2}10\right)}^{-1}-x$

$\left({\log }_{10}8\right)x^2-\left({\log }_{10}5\right)x=\frac{2}{{\log }_{2}10}-x$

$\Rightarrow \left({\log }_{10}8\right)x^2-\left(1-{\log }_{10}5\right)x-\frac{2}{{\log }_{2}10}=0$

$\Rightarrow \left({\log }_{10}8\right)x^2-\left[{\log }_{10}\left(\frac{10}{5}\right)\right]x-2{\log }_{10}2=0$

$\Rightarrow \left({\log }_{10}8\right)x^2-\left({\log }_{10}2\right)x-{\log }_{10}4=0$

Sum of the roots $=\frac{-b}{a}=\frac{{\log }_{10}2}{{\log }_{10}8}={\log }_{8}2=\frac{1}{3}$

Product of the roots $=\frac{c}{a}=\frac{{\log }_{10}4}{{\log }_{10}8}=\frac{-2}{3}$

Discriminant $\Rightarrow b^2-4ac$

$\Rightarrow {\left({\log }_{10}2\right)}^2+4.{\log }_{10}8{\log }_{10}4$

$=\left({\log }_{10}2\right)+4\times 3\times 2{\left({\log }_{10}2\right)}^2$

$=25{\left({\log }_{10}2\right)}^2$

Sum of the coefficients $={\log }_{10}8-{\log }_{10}4-{\log }_{10}2$

$=3{\log }_{10}2-2{\log }_{10}2-{\log }_{10}2$