tex]...

August 2022 1 26 Report

Решите уравнение методом замены переменной. Укажите все положительные корни уравнения.
[tex](x^{2} - 2x - 3)^{2} = x^{2} - 2x + 17[/tex]

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SYSTEMCORE

$\displaystyle (x^2-2x-3)^2=x^2-2x+17\\\\(x^2-2x-3)^2=x^2-2x-3+20\\\\x^2-2x-3=t\\\\t^2=t+20\\\\t^2-t-20=0\\\\\text{D}=1+4\cdot1\cdot20=81=9^2\\\\t_1=\frac{1+9}2=\frac{10}2=5\\\\t_2=\frac{1-9}2=\frac{-8}2=-4$

$x^2-2x-3=t, \quad t=5\\\\x^2-2x-3=5\\\\x^2-2x-8=0\\\\\text{D}=4+4\cdot1\cdot8=36=6^2\\\\x_1=\frac{2+6}{2}=\boxed{4}\\\\x_2=\frac{2-6}2=-\frac{4}2=-2\\\\\\x^2-2x-3=t,\quad t=-4\\\\x^2-2x-3=-4\\\\x^2-2x+1=0\\\\(x-1)^2=0\\\\\boxed{x=1}$

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