Solve the Following Quadratic Equation by Factorization
Question:
\[ \frac{x-1}{2x+1}+\frac{2x+1}{x-1}=\frac{5}{2}, \qquad x\ne -\frac12,\;1 \]Solution
Given:
\[ \frac{x-1}{2x+1}+\frac{2x+1}{x-1}=\frac{5}{2} \]Taking LCM on the left side:
\[ \frac{(x-1)^2+(2x+1)^2}{(2x+1)(x-1)} =\frac{5}{2} \]Cross-multiplying:
\[ 2\Big[(x-1)^2+(2x+1)^2\Big] =5(2x+1)(x-1) \] \[ 2\Big[(x^2-2x+1)+(4x^2+4x+1)\Big] =5(2x^2-x-1) \] \[ 10x^2+4x+4 =10x^2-5x-5 \] \[ 9x+9=0 \] \[ x+1=0 \] \[ x=-1 \]Since \(x=-1\) does not violate the restrictions \(x\ne -\frac12,1\), it is a valid solution.