How do I use the quadratic formula to solve $\frac{1}{x + 1} - \frac{1}{x} = \frac{1}{2}$ $1/(x+1) - 1/x = 1/2$ ?

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Answer 1 · 2018-04-07T03:44:51

$x_{+} = \frac{- 1 + i \sqrt{7}}{2}, x_{- =} \frac{- 1 - i \sqrt{7}}{2}$ $color(green)(x_+ = (-1 + isqrt7)/2, " " x_- = (-1 - isqrt7)/2$

$(\frac{1}{x + 1}) - \frac{1}{x} = \frac{1}{2}$ $(1/(x+1)) - 1/x = 1/2$

$\frac{x - x - 1}{x \cdot (x + 1)} = \frac{1}{2}, taking L C M for L H S$ $(x - x - 1) / (x * (x+1)) = 1/2, " taking L C M for L H S"$

$- 1 \cdot 2 = x \cdot (x + 1), cross multiplying$ $-1 * 2 = x * (x + 1), " cross multiplying"$

$x^{2} + x = - 2$ $x^2 + x = - 2$

$x^{2} + x + 2 = 0$ $x^2 + x + 2 = 0$

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$Applying the above formula with$ $"Applying the above formula with "$ a = 1, b = 1, c = 2 $for finding the roots$ $" for finding the roots"$ ,

$x_{+} = \frac{- 1 + (\sqrt{1 - 8})}{2} = \frac{- 1 + i \sqrt{7}}{2}$ $x_+ = (-1 + (sqrt(1-8))) / 2 = (-1 + isqrt7)/2$

$x_{- =} \frac{- 1 - (\sqrt{1 - 8})}{2} = \frac{- 1 - i \sqrt{7}}{2}$ $x_- = (-1 - (sqrt(1-8))) / 2 = (-1 - isqrt7)/2$

How do I use the quadratic formula to solve 1/(x+1) - 1/x = 1/21x+1−1x=121/(x+1) - 1/x = 1/2?