Simplify equation: $\frac{1}{x} - \frac{1}{(x + 1)} = \frac{1}{(x + 2)} - \frac{1}{(x + 4)}$ to get a quadratic equation. Find the nature of roots. Solve the equation using the formula.

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Solution

$\to$ The following eq can be simplified to,
$\to$ $\frac{1}{(x) (x + 1)} = \frac{2}{(x + 2) (x + 4)}$
by cross multipilcation method we are getting
$\to$ $x^{2} + 8 + 6 x = 2 x^{2} + 2 x$
further solving,
$\to$ $x^{2} - 4 x - 8 = 0$
now finding D of the given eq,
$\to$ $D = b^{2} - 4 a c$
$\to$ $D = 16 - 4 \times (- 8) \times 1$
$\to$ $D = 48$ Since the D is positive there The eq has the real and distinct roots now solving eq
$\to$ $x = (- b + D^{1 / 2}) / 2 a$
$\to$ $x = (- b - D^{1 / 2}) / 2 a$
solving these eq, we get
$\to$ $x = 0.341$ , and $x = 0.09150$

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