Простейшие показательные уравнения. Задача 25

ОТВЕТ: 4.

\({2^{2x-3}}-{4^{{x^2}-3x-1}} = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{2^{2x-3}} = {2^{2{x^2}-6x-2}}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,2x-3 = 2{x^2}-6x-2\,\,\,\,\,\,\, \Leftrightarrow \)

\( \Leftrightarrow \,\,\,\,\,\,\,2{x^2}-8x + 1 = 0;\,\,\,\,\,\,\,\,\,\,\,D = {\left( {-8} \right)^2}-4 \cdot 2 \cdot 1 = 64-8 = 56 > 0.\)

По теореме Виета:   \({x_1} + {x_2} = -\dfrac{b}{a} = -\dfrac{{-8}}{2} = 4.\)

Ответ:  4.