Solve the Following Quadratic Equation by Factorization
Question:
\[ \frac{3}{x+1}-\frac{1}{2}=\frac{2}{3x-1}, \qquad x\ne -1,\frac{1}{3} \]Solution
Given:
\[ \frac{3}{x+1}-\frac{1}{2}=\frac{2}{3x-1} \]Multiplying both sides by \(2(x+1)(3x-1)\):
\[ 6(3x-1)-(x+1)(3x-1)=4(x+1) \] \[ 18x-6-(3x^2+2x-1)=4x+4 \] \[ -3x^2+16x-5=4x+4 \] \[ -3x^2+12x-9=0 \] \[ 3x^2-12x+9=0 \] \[ x^2-4x+3=0 \]Factorizing:
\[ x^2-3x-x+3=0 \] \[ x(x-3)-1(x-3)=0 \] \[ (x-3)(x-1)=0 \]Therefore,
\[ x-3=0 \quad \text{or} \quad x-1=0 \] \[ x=3 \quad \text{or} \quad x=1 \]Both values satisfy the condition \(x\ne -1,\frac{1}{3}\).