Mathematics - Quadratic Equation Question with Solution | TestHub

If $a_1{x}^2+b_{1}x+c_1=0$ and $a_2{x}^2+b_{2}x+c_2=0$ have a common real root, then $$\begin{array}{rl}\Rightarrow {\left(a_1{c}_2-a_2{c}_1\right)}^2=(b_1{c}_2-& b_2{c}_1)\\ & \left(a_1{b}_2-a_2{b}_1\right)\end{array}$$ $\therefore \begin{array}{l}{x}^2+bx-1=0\\ x^2+x+b=0\end{array}\}$ have a common root. $\Rightarrow (1+b{)}^2=\left(b^2+1\right)(1-b)$ $\Rightarrow b^2+2b+1=b^2-b^3+1-b$ $\Rightarrow b^3+3b=0\Rightarrow b\left(b^2+3\right)=0$ $\Rightarrow b=0,\pm \sqrt{3}i$