Solve the Following Quadratic Equation by Factorization
Question:
\[ \frac{x-a}{x-b}+\frac{x-b}{x-a} = \frac{a}{b}+\frac{b}{a} \]Solution
Given:
\[ \frac{x-a}{x-b}+\frac{x-b}{x-a} = \frac{a}{b}+\frac{b}{a} \]Taking LCM on both sides:
\[ \frac{(x-a)^2+(x-b)^2}{(x-a)(x-b)} = \frac{a^2+b^2}{ab} \]Cross-multiplying:
\[ ab\left[(x-a)^2+(x-b)^2\right] = (a^2+b^2)(x-a)(x-b) \]Expanding and simplifying:
\[ 2abx^2-2ab(a+b)x+ab(a^2+b^2) = (a^2+b^2)\left[x^2-(a+b)x+ab\right] \] \[ (a-b)^2x^2-(a-b)^2(a+b)x=0 \] \[ (a-b)^2x\,[x-(a+b)]=0 \]Therefore,
\[ x=0 \] or \[ x=a+b \]