Question

Find the value of the expression​

Answer 1

➫Question :-

$\rm \: The \: value \: of \: \\ \to \: \blue{ \sqrt{8 + 2 \sqrt{ 8 + 2 \sqrt{8 + 2...... \infty} } } }$

➫Answer :-

$\green{ \boxed{ \red{ \sqrt{8 + 2 \sqrt{ 8 + 2 \sqrt{8 + 2...... \infty} } } } = 4}} \:$

➫Step - by - step explanation :-

We have ,

$\mapsto \: \sqrt{8 + 2 \sqrt{ 8 + 2 \sqrt{8 + 2...... \infty} } } \: \\ \\ \sf \: let \: \\ \mapsto \: x = \sqrt{8 + 2 \sqrt{ 8 + 2 \sqrt{8 + 2...... \infty} } } \: \\ \\ \sf \green{ we \: can \: write \: it }\: \\ \\ \mapsto \: x = \sqrt{8 + 2 { x} } \\ \\ \rm \red{ squaring \: on \: both \: sides} \\ \\ \mapsto \: {x}^{2} + { \bigg \{\sqrt{8 + 2 {x} } \bigg \} }^{2} \\ \\ \mapsto \: {x}^{2} = 8 + 2 {x} \\ \\ \mapsto \: {x}^{2} - 2x - 8 = 0 \\ \\ \rm \red{ solve \: this \: quadratic }\: by \: splitting \: \\ \rm \pink{ the \: middle} \: term \\ \\ \mapsto \: {x}^{2} - (4 - 2)x - 8 = 0 \\ \\ \mapsto \: {x}^{2} - 4x + 2x - 8 = 0 \\ \\ \mapsto \: x(x - 4) + 2(x - 4) = 0 \\ \\ \mapsto \: (x - 4)(x + 2) = 0 \\ \\ \star \sf \: case \: (1) \: if \\ \\ \to \: x - 4 = 0 \\ \\ \to \: \boxed{x = 4} \\ \\ \star \rm \: case \: (2) \: if \\ \\ \to \: x + 2 = 0 \\ \\ \to \: x = - 2 \\ \\ \rm this \: sum\: of \: \: numbers \: may \: or \: may \: not \\ \rm \: be \: negative \: \\ \sf hence \: \: \\ \to \: \boxed{ \: x = 4} \\ \\ \green{ \boxed{ \red{ \sqrt{8 + 2 \sqrt{ 8 + 2 \sqrt{8 + 2...... \infty} } } } = 4}}$