Solve the Following Quadratic Equation by Factorization
Question:
\[ \frac{1}{2a+b+2x} = \frac{1}{2a} + \frac{1}{b} + \frac{1}{2x} \]Solution
Given,
\[ \frac{1}{2a+b+2x} = \frac{1}{2a} + \frac{1}{b} + \frac{1}{2x} \]Taking LCM on the right-hand side:
\[ \frac{bx+2ax+ab} {2abx} = \frac{1} {2a+b+2x} \]Cross-multiplying:
\[ 2abx = (2a+b+2x)(bx+2ax+ab) \]Since
\[ bx+2ax+ab=a(2x+b)+bx \] \[ =(a+x)(2a+b) \]Therefore,
\[ 2abx=(2a+b+2x)(a+x)(2a+b) \]Expanding and simplifying gives
\[ (2x+a)(2x+b)=0 \]Hence,
\[ 2x+a=0 \] or \[ 2x+b=0 \] \[ x=-\frac{a}{2} \] or \[ x=-\frac{b}{2} \]