Question

Question) Solve the following equation for x by factorization :
$a^2{x}^2+(a^2+b^2)x+b^2=0;a\ne 0$

Answer 1

$$\begin{gathered} {\mathrm{a}}^2{\mathrm{x}}^2+\left({\mathrm{a}}^2+{\mathrm{b}}^2\right)\mathrm{x}+{\mathrm{b}}^2=0 \\ \Rightarrow {\mathrm{a}}^2{\mathrm{x}}^2+{\mathrm{a}}^2\mathrm{x}+{\mathrm{b}}^2\mathrm{x}+{\mathrm{b}}^2=0 \\ \Rightarrow {\mathrm{a}}^2\mathrm{x}\left(\mathrm{x}+1\right)+{\mathrm{b}}^2\left(\mathrm{x}+1\right)=0 \\ \Rightarrow \left({\mathrm{a}}^2\mathrm{x}+{\mathrm{b}}^2\right)\left(\mathrm{x}+1\right)=0 \\ \Rightarrow {\mathrm{a}}^2\mathrm{x}+{\mathrm{b}}^2=0\mathrm{or}\mathrm{x}+1=0 \\ \Rightarrow \mathrm{x}=-\frac{{\mathrm{b}}^2}{{\mathrm{a}}^2}\mathrm{or}\mathrm{x}=-1 \end{gathered}$$