Question

Solve the following equation and verify your answer:
\[\left( \frac{x + 1}{x - 4} \right)^2 = \frac{x + 8}{x - 2}\]

Answer 1

\[ \left( \frac{x + 1}{x - 4} \right)^2 = \frac{x + 8}{x - 2}\]

\[\text{ or }\frac{x^2 + 2x + 1}{x^2 - 8x + 16} = \frac{x + 8}{x - 2} [(a + b )^2 = a^2 + b^2 + 2ab\text{ and }(a - b )^2 = a^2 + b^2 - 2ab ]\]

\[\text{ or }x^3 + 2 x^2 + x - 2 x^2 - 4x - 2 = x^3 - 8 x^2 + 16x + 8 x^2 - 64x + 128 [\text{ After cross multiplication }]\]

\[\text{ or }x^3 - x^3 - 3x + 48x = 128 + 2\]

\[\text{ or }45x = 130\]

\[\text{ or }x = \frac{130}{45} = \frac{26}{9}\]

\[\text{ Thus }x = \frac{26}{9}\text{ is the solution of the given equation .} \]

\[\text{ Check: }\]

\[\text{ Substituting }x = \frac{26}{9} \text{ in the given equation, we get: }\]

\[\text{ L . H . S . }= \left( \frac{\frac{26}{9} + 1}{\frac{26}{9} - 4} \right)^2 = \left( \frac{26 + 9}{26 - 36} \right)^2 = \frac{1225}{100} = \frac{49}{4}\]

\[\text{ R . H . S .} = \left( \frac{\frac{26}{9} + 8}{\frac{26}{9} - 2} \right) = \left( \frac{26 + 72}{26 - 18} \right) = \frac{98}{8} = \frac{49}{4}\]

\[ \therefore\text{ L . H . S . = R . H . S . for }x = \frac{26}{9}\]