Question

$\begin{array}{c}{x}^2+abx+c=0\\ x^2+acx+b=0\end{array}\}$ one common root $\alpha$.
Find the quadratic equation containing their common root

Answer 1

$\begin{array}{c}{x}^2+abx+c=0\\ x^2+acx+b=0\\ \left(ab-ac\right)x+\left(c-b\right)=0\\ a\left(b-c\right)x=b-c\\ ax=1\\ x=\frac{1}{a}\\ \alpha =\frac{1}{a}\\ {\left(\frac{1}{a}\right)}^2+ab\times \frac{1}{a}+c=0\\ \frac{1}{a^2}+b+c=0\\ 1+a^{2}b+c{a}^2=0\\ a^2\left(c+b\right)+1=0\end{array}$