Question

If the equations x² + x + a = 0 and x² + ax + 1 = 0, a ≠ 1, have a common root, then a = ____________.

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ {x}^2+x+a=0\mathrm{and} \\ {x}^2+ax+1=0\mathrm{have}\mathrm{a}\mathrm{common}\mathrm{root}\mathrm{where}\mathit{}a\ne 1. \\ \mathrm{Solving}\mathrm{above}2\mathrm{equations}, \\ \mathrm{we}\mathrm{get} \\ \underline{\begin{array}{c}\begin{array}{cccccccc}& x^2& +& x& +& a& =& 0\\ & x^2& +& ax& +& 1& =& 0\\ -& & -& & -& & & \end{array}\end{array}} \\ x\left(1-a\right)+a-1=0 \\ \mathrm{i}.\mathrm{e}.x\left(1-a\right)=+1-a \\ \mathrm{i}.\mathrm{e}.x=1\left(\because a\ne 1\right)\mathrm{given} \\ \therefore a\mathrm{can}\mathrm{be}\mathrm{obtained}\mathrm{form}{x}^2+x+a=0 \\ \mathrm{Since}x=1\mathrm{satisfy}{x}^2+x+a=0 \\ \mathrm{i}.\mathrm{e}.{\left(1\right)}^2+1+a=0 \\ \mathrm{i}.\mathrm{e}.a=-2 \end{gathered}$$