The sum of the roots of the equation [x + 1 - 2 _2(3 + 2^x) + 2 _4(10 - 2^ - x) = 0], is :

Question

The sum of the roots of the equation $$x + 1 - 2{\log _2}(3 + {2^x}) + 2{\log _4}(10 - {2^{ - x}}) = 0$$, is :

Accepted Answer

$$x + 1 - 2{\log _2}(3 + {2^x}) + 2{\log _4}(10 - {2^{ - x}}) = 0$$

$${\log _2}({2^{x + 1}}) - {\log _2}{(3 + {2^x})^2} + {\log _2}(10 - {2^{ - x}}) = 0$$

$$lo{g_2}\left( {{{{2^{x + 1}}.(10 - {2^{ - x}})} \over {{{(3 + {2^x})}^2}}}} \right) = 0$$

$${{2({{10.2}^{ - x}} - 1)} \over {{{(3 + {2^x})}^2}}} = 1$$

$$ \Rightarrow {20.2^x} - 2 = 9 + {2^{2x}} + {6.2^x}$$

$$\therefore$$ ${({2^x})^2} - 14({2^x}) + 11 = 0$

Roots are 2x1 \& 2x2

$$\therefore$$ 2x1 . 2x2 = 11

x1 + x2 = log2(11)

Solutions · 1