Question

Solve the following quadratic equations by factorization:

\[\frac{4}{x} - 3 = \frac{5}{2x + 3}, x \neq 0, - \frac{3}{2}\]

Answer 1

\[\frac{4}{x} - 3 = \frac{5}{2x + 3}\]

\[ \Rightarrow \frac{4 - 3x}{x} = \frac{5}{2x + 3}\]

\[ \Rightarrow \left( 4 - 3x \right)\left( 2x + 3 \right) = 5x\]

\[ \Rightarrow 8x + 12 - 6 x^2 - 9x = 5x\]

\[ \Rightarrow - 6 x^2 - 6x + 12 = 0\]

\[ \Rightarrow x^2 + x - 2 = 0\]

\[ \Rightarrow x^2 + 2x - x - 2 = 0\]

\[ \Rightarrow x(x + 2) - 1(x + 2) = 0\]

\[ \Rightarrow (x - 1)(x + 2) = 0\]

\[ \Rightarrow x - 1 = 0 \text { or } x + 2 = 0\]

\[ \Rightarrow x = 1 \text { or } x = - 2\]

Hence, the factors are 1 and −2.