Question

6m² + m = 2

Answer 1

Given: 6m² + m = 2
On dividing both sides by 6, we get:
$m^2+\frac{m}{6}=\frac{1}{3}$ …(1)
We make L.H.S. a perfect square by using
Third term = ${\left(\frac{1}{2}\times \mathrm{coefficient}\mathrm{of}m\right)}^2={\left(\frac{1}{2}\times \frac{1}{6}\right)}^2={\left(\frac{1}{12}\right)}^2=\frac{1}{144}$
On adding 1/144 to both sides of equation (1), we get:
$$\begin{gathered} m^2+\frac{m}{6}+\frac{1}{144}=\frac{1}{3}+\frac{1}{144} \\ \Rightarrow m^2+2\times m\times \frac{1}{12}+{\left(\frac{1}{12}\right)}^2=\frac{48+1}{144} \\ \Rightarrow {\left(m+\frac{1}{12}\right)}^2=\frac{49}{144} \\ \Rightarrow {\left(m+\frac{1}{12}\right)}^2={\left(\frac{7}{12}\right)}^2 \end{gathered}$$
On taking square root of both sides, we get:
$$\begin{gathered} m+\frac{1}{12}=\pm \frac{7}{12} \\ \Rightarrow m=\frac{-1}{12}\pm \frac{7}{12} \\ \Rightarrow m=\frac{-1}{12}+\frac{7}{12},\frac{-1}{12}-\frac{7}{12} \\ =\frac{-1+7}{12},\frac{-1-7}{12} \\ =\frac{6}{12},\frac{-8}{12} \\ \Rightarrow m=\frac{1}{2},\frac{-2}{3} \end{gathered}$$