Solve the Following Quadratic Equation by Factorization
Question:
\[ \frac{3}{x+1}+\frac{4}{x-1}=\frac{29}{4x-1}, \qquad x\ne 1,-1,\frac14 \]Solution
Given:
\[ \frac{3}{x+1}+\frac{4}{x-1}=\frac{29}{4x-1} \]Multiplying both sides by \((x+1)(x-1)(4x-1)\):
\[ 3(x-1)(4x-1)+4(x+1)(4x-1) =29(x+1)(x-1) \] \[ 3(4x^2-5x+1)+4(4x^2+3x-1) =29(x^2-1) \] \[ 12x^2-15x+3+16x^2+12x-4 =29x^2-29 \] \[ 28x^2-3x-1 =29x^2-29 \] \[ x^2+3x-28=0 \]Factorizing:
\[ x^2+7x-4x-28=0 \] \[ x(x+7)-4(x+7)=0 \] \[ (x+7)(x-4)=0 \]Therefore,
\[ x+7=0 \quad \text{or} \quad x-4=0 \] \[ x=-7 \quad \text{or} \quad x=4 \]Both values satisfy the condition \(x\ne1,-1,\frac14\).