Ответ: x₁ = 4 , x₂ = -2
Пошаговое объяснение:
[tex](x^2 -2x )^2 - (x-1)^2=55 \\\\ (x (x-2))^2 - (x-1)^2 = 55[/tex]
Сделаем замену [tex]t = x-1 \\\\ t -1 = x-2 \\\\ t + 1 = x[/tex]
[tex]\Big ( ( t -1)(t+1) \Big ) ^2 - t^2 = 55 \\\\ (t ^2-1)^2 -t^2 = 55 \\\\ t^4 - 2t^2 +1 -t^2 -55 =0 \\\\ t^4 -3t^2 -54 =0[/tex]
Выходит биквадратное уравнение
[tex]u = t^2 ~~ , ~~ u^2 = t^4 ~ , ~ u > 0 \\\\ u^2-3u -54 = 0 \\\\ \displaystyle \left \{ {{u_1 +u_2=3} \atop {u_1 u_2 = -54}} \right. \Leftrightarrow u_1 = 9 ~\checkmark ~~ , ~~ u_2 = - 6 ~~ \varnothing[/tex]
Тогда [tex]t^2 = 9 \\\\ t _{1,2} = \pm 3[/tex]Вернемся к старой переменой [tex]t = x-1[/tex]
[tex]\hspace{-1,4em }1. ~~ x- 1 = 3 \\\\ x_1= 4 \\\\[/tex]
[tex]\hspace{-1,4em }2. ~~ x- 1 = -3 \\\\ x_2= -2\\\\[/tex]
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Verified answer
Ответ: x₁ = 4 , x₂ = -2
Пошаговое объяснение:
[tex](x^2 -2x )^2 - (x-1)^2=55 \\\\ (x (x-2))^2 - (x-1)^2 = 55[/tex]
Сделаем замену
[tex]t = x-1 \\\\ t -1 = x-2 \\\\ t + 1 = x[/tex]
[tex]\Big ( ( t -1)(t+1) \Big ) ^2 - t^2 = 55 \\\\ (t ^2-1)^2 -t^2 = 55 \\\\ t^4 - 2t^2 +1 -t^2 -55 =0 \\\\ t^4 -3t^2 -54 =0[/tex]
Выходит биквадратное уравнение
[tex]u = t^2 ~~ , ~~ u^2 = t^4 ~ , ~ u > 0 \\\\ u^2-3u -54 = 0 \\\\ \displaystyle \left \{ {{u_1 +u_2=3} \atop {u_1 u_2 = -54}} \right. \Leftrightarrow u_1 = 9 ~\checkmark ~~ , ~~ u_2 = - 6 ~~ \varnothing[/tex]
Тогда
[tex]t^2 = 9 \\\\ t _{1,2} = \pm 3[/tex]
Вернемся к старой переменой [tex]t = x-1[/tex]
[tex]\hspace{-1,4em }1. ~~ x- 1 = 3 \\\\ x_1= 4 \\\\[/tex]
[tex]\hspace{-1,4em }2. ~~ x- 1 = -3 \\\\ x_2= -2\\\\[/tex]