Solve the following equations :

$(x - 5) (x + 3) = (x - 7) (x + 4)$

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Solution

$(x - 5) (x + 3) = (x - 7) (x + 4) \Rightarrow x^{2} + 3 x - 5 x - 15 = x^{2} + 4 x - 7 x - 28 \Rightarrow - 2 x - 15 = - 3 x - 28 \Rightarrow 3 x - 2 x = 15 - 28 \Rightarrow x = - 13$

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(x - 5) (x + 3) = (x - 7) (x + 4)

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\frac{2 - 7 x}{1 - 5 x} = \frac{3 + 7 x}{4 + 5 x}

\frac{3 x^{4} + x^{2} - 2 x - 3}{3 x^{4} - x^{2} + 2 x + 3} = \frac{5 x^{4} + 2 x^{2} - 7 x + 3}{5 x^{4} - 2 x^{2} + 7 x - 3}

\frac{5 x - 2}{3} - \frac{7 x - 3}{5} > \frac{x}{4}

(x - 7) (x - 3) (x + 5) (x + 1) = 1680

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