Question

solve for x : 8√(x/x+3) - √(x+3/x) = 2​

Answer 1

$\large\underline\purple{\bold{Solution :- }}$

$\tt \ \: :  ⟼ 8 \sqrt{\dfrac{x}{x + 3} } - \sqrt{\dfrac{x + 3}{x} } = 2$

$\tt \ \: :  ⟼ Let \: \sqrt{\dfrac{x}{x + 3} } = y$

$\tt\implies \:8y - \dfrac{1}{y} = 2$

$\tt\implies \:\dfrac{ {8y}^{2} - 1}{y} = 2$

$\tt\implies \: {8y}^{2} - 1 = 2y$

$\tt\implies \: {8y}^{2} - 2y - 1 = 0$

$\tt\implies \: {8y}^{2} - 4y + 2y - 1 = 0$

$\tt\implies \:4y(2y - 1) + 1(2y - 1) = 0$

$\tt\implies \:(4y + 1)(2y - 1) = 0$

$\tt\implies \:y = - \dfrac{1}{4} \: or \: y = \dfrac{1}{2}$

$\tt\implies \:\sqrt{\dfrac{x}{x + 3} } = - \dfrac{1}{4} \: or \: \sqrt{\dfrac{x}{x + 3} } =\dfrac{1}{2}$

$\tt\implies \:\dfrac{x}{x + 3 } = \dfrac{1}{16} \: or \: \dfrac{x}{x + 3} = \dfrac{1}{4}$

$\tt\implies \:16x = x + 3 \: or \: 4x = x + 3$

$\tt\implies \:15x = 3 \: or \: 3x = 3$

$\tt\implies \:x = \dfrac{1}{5} \: or \: x \: = 1$