Solve the Following Quadratic Equation by Factorization
Question:
\[ \frac{x+3}{x-2}-\frac{1-x}{x}=\frac{17}{4}, \qquad x\ne 0,2 \]Solution
Given:
\[ \frac{x+3}{x-2}-\frac{1-x}{x}=\frac{17}{4} \]Since \(1-x=-(x-1)\), we have
\[ \frac{x+3}{x-2}+\frac{x-1}{x}=\frac{17}{4} \]Multiplying both sides by \(4x(x-2)\):
\[ 4x(x+3)+4(x-1)(x-2)=17x(x-2) \] \[ 4x^2+12x+4(x^2-3x+2)=17x^2-34x \] \[ 8x^2+8+17x^2-34x \] \[ 8x^2+8=17x^2-34x \] \[ 9x^2-34x-8=0 \]Factorizing:
\[ 9x^2-36x+2x-8=0 \] \[ 9x(x-4)+2(x-4)=0 \] \[ (9x+2)(x-4)=0 \]Therefore,
\[ 9x+2=0 \quad \text{or} \quad x-4=0 \] \[ x=-\frac{2}{9} \quad \text{or} \quad x=4 \]Both values satisfy the condition \(x\ne0,2\).