Solve the Following Quadratic Equation by Factorization
Question:
\[ x^2+\left(a+\frac1a\right)x+1=0 \]Solution
Given,
\[ x^2+\left(a+\frac1a\right)x+1=0 \]Splitting the middle term:
\[ x^2+ax+\frac{x}{a}+1=0 \]Taking common factors:
\[ x(x+a)+\frac1a(x+a)=0 \] \[ (x+a)\left(x+\frac1a\right)=0 \]Therefore,
\[ x+a=0 \] or \[ x+\frac1a=0 \] \[ x=-a \] or \[ x=-\frac1a \]