Question

Solve: √x/1+x + √1+x/x = 5/2​

Answer 1

Step-by-step explanation:

$\implies \: \sf\sqrt{ \dfrac{x}{1 + x} } + \sqrt{ \dfrac{1 + x}{x} } = \dfrac{5}{2}$

Do squaring on both sides,

$\implies \sf { \big(\sqrt{ \dfrac{x}{1 + x} }\big)}^{2} + { \big(\sqrt{ \dfrac{1 + x}{x} } \big)}^{2} = { \big(\dfrac{5}{2} \big)}^{2}$

$\implies \:\sf\dfrac{x}{1 + x} + \dfrac{1 + x}{x} = \dfrac{25}{4}$

Take LCM

$\implies \sf\dfrac{x(x) + (1 + x)(1 + x)}{x(1 + x)} = \dfrac{25}{4}$

$\implies \sf \dfrac{ {x}^{2} + 1(1 + x) + x(1 + x)}{x + {x}^{2} } = \dfrac{25}{4}$

$\implies \sf \dfrac{ {x}^{2} + 1 + x + x + {x}^{2} }{x + {x}^{2} } = \dfrac{25}{4}$

$\implies \sf \dfrac{2 {x}^{2} + 2x + 1 }{ {x}^{2} + x} = \dfrac{25}{4}$

$\implies \sf4(2 {x}^{2} + 2x + 1) = 25( {x}^{2} + x)$

$\implies \sf8 {x}^{2} + 8x + 4 = 25 {x}^{2} + 25x$

$\implies \sf25 {x}^{2} - 8 {x}^{2} + 25x - 8x - 4 = 0$

$\implies \sf17 {x}^{2} + 17x - 4 = 0$

On solving the quadratic equation 17x² + 17x - 4 = 0 using formula: x = [-b ± √(b² - 4ac)]/2a where the value of a is 17, b is 17 and c is - 4, we get,

$\implies \sf \: x = \frac{ - 17 + \sqrt{561} }{34} \: and \: x = \frac{ - 17 - \sqrt{561} }{34}$