Solve the Following Quadratic Equation by Factorization
Question:
\[ a(x^2+1)-x(a^2+1)=0 \]Solution
Given,
\[ a(x^2+1)-x(a^2+1)=0 \]Expanding,
\[ ax^2+a-a^2x-x=0 \] \[ ax^2-(a^2+1)x+a=0 \]Splitting the middle term:
\[ ax^2-a^2x-x+a=0 \]Taking common factors:
\[ ax(x-a)-1(x-a)=0 \] \[ (x-a)(ax-1)=0 \]Therefore,
\[ x-a=0 \] or \[ ax-1=0 \] \[ x=a \] or \[ x=\frac{1}{a} \]