Question

Solve the following quadratic equations by factorization: \[\frac{5 + x}{5 - x} - \frac{5 - x}{5 + x} = 3\frac{3}{4}; x \neq 5, - 5\]

Answer 1

\[\frac{5 + x}{5 - x} - \frac{5 - x}{5 + x} = 3\frac{3}{4}\]

\[ \Rightarrow \frac{\left( 5 + x \right)^2 - \left( 5 - x \right)^2}{\left( 5 + x \right)\left( 5 - x \right)} = \frac{15}{4}\]

\[ \Rightarrow \frac{25 + x^2 + 10x - 25 - x^2 + 10x}{25 - x^2} = \frac{15}{4}\]

\[ \Rightarrow \frac{20x}{25 - x^2} = \frac{15}{4}\]

\[ \Rightarrow \frac{4x}{25 - x^2} = \frac{3}{4}\]

\[ \Rightarrow 16x = 75 - 3 x^2 \]

\[ \Rightarrow 3 x^2 + 16x - 75 = 0\]

\[ \Rightarrow 3 x^2 + 25x - 9x - 75 = 0\]

\[ \Rightarrow x(3x + 25) - 3(3x + 25) = 0\]

\[ \Rightarrow (x - 3)(3x + 25) = 0\]

\[ \Rightarrow x - 3 = 0 \text { or } 3x + 25 = 0\]

\[ \Rightarrow x = 3 \text { or } x = - \frac{25}{3}\]

Hence, the factors are 3 and \[- \frac{25}{3}\].