Question

solve for x : x/x-1 + x-1/x =4; x is not equal to 0 and 1

Answer 1

Step-by-step explanation:

$\small{ = > \frac{x}{x -1 } + \frac{x - 1}{x} = 4 } \\ \\ \small{ = > \frac{{x}^{2} + (x - 1) ^{2} }{x(x - 1)} } = 4$

= > x² + (x² + 1² - 2x) = 4x(x - 1)

= > x² + x² + 1 - 2x = 4x² - 4x

= > 2x² + 1 - 2x = 4x² - 4x

= > 2x² - 4x² + 1 - 2x + 4x = 0

= > - 2x² + 1 + 2x = 0

= > 2x² - 1 - 2x = 0

= > 2x² - 2x - 1 = 0

Using quadratic formula:

$\small{ = > x = \frac{ - (-2) \pm \sqrt{2 {}^{2} - 4( - 1)(2)} }{2(2)} } \\ \\ \small{ = >x = \frac{ 2 \pm \sqrt{4 + 8} }{4} }\\ \\ \small{ = >x = \frac{ 2 \pm \sqrt{12} }{4} }\\ \\ \small{ = >x = \frac{ 2 \pm \sqrt{4 \times 3} }{4} = \frac{ 2 \pm2 \sqrt{3} }{4} } \\ \\ \small{ = >x = \frac{ 1\pm \sqrt{3} }{2} }$