Question

$x = \sqrt{5 + \sqrt{5 + \sqrt{5 + ..... \infty } } }$

Answer 1

★ QUADRATIC RESOLUTIONS ★

$x = \sqrt{5 + \sqrt{5 + \sqrt{5 + \sqrt{5 + .... \infty } } } } \\ \\ x = \sqrt{5 + x} \\ {x}^{2} = 5 + x \\ \\ {x}^{2} - x - 5 = 0 \\ d = {b}^{2} - 4ac \: = { (- 1)}^{2} - 4(1)( - 5) = 21\\ \\ x = \frac{ - b + \sqrt{ {b}^{2} - 4ac } }{2a} \: or \: \\ x = \frac{ - b - \sqrt{ {b}^{2} - 4ac } }{2a} \\ \\ x = \frac{1 - \sqrt{21} }{2} \: or \\ \\ x = \frac{1 + \sqrt{21} }{2}$