Solve the Following Quadratic Equation by Factorization
Question:
\[ \frac{2}{x+1}+\frac{3}{2(x-2)}=\frac{23}{5x}, \qquad x\ne 0,-1,2 \]Solution
Given:
\[ \frac{2}{x+1}+\frac{3}{2(x-2)}=\frac{23}{5x} \]Multiplying both sides by \(10x(x+1)(x-2)\):
\[ 20x(x-2)+15x(x+1) =46(x+1)(x-2) \] \[ 20x^2-40x+15x^2+15x =46(x^2-x-2) \] \[ 35x^2-25x =46x^2-46x-92 \] \[ 11x^2-21x-92=0 \]Factorizing:
\[ 11x^2-44x+23x-92=0 \] \[ 11x(x-4)+23(x-4)=0 \] \[ (x-4)(11x+23)=0 \]Therefore,
\[ x-4=0 \quad \text{or} \quad 11x+23=0 \] \[ x=4 \quad \text{or} \quad x=-\frac{23}{11} \]Both values satisfy the condition \(x\ne0,-1,2\).