Задача 71907 3*81^x - 10*9^x + 3 = 0 2^x + 2^(2-x) =...

64099668438f4e0b023ada83

01.06.2023 10:44:02

3*81^x - 10*9^x + 3 = 0

2^x + 2^(2-x) = 5

626fab9a354ab65fb2f43abb

01.06.2023 15:04:08

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1 задача.
3*81^(x) - 10*9^(x) + 3 = 0
Замена y = 9^(x) > 0 при любом x; тогда 81^(x) = y^2
3y^2 - 10y + 3 = 0
D = (-10)^2 - 4*3*3 = 100 - 36 = 64 = 8^2
y1 = 9^(x) = (10 - 8)/6 = -2/6 = -1/3 < 0 - не подходит.
y2 = 9^(x) = (10 + 8)/6 = 18/6 = 3; x2 = 1/2 - подходит

Ответ: x = 1/2

2 Задача.
2^(x) + 2^(2 - x) = 5
2^(x) + 2^2/2^(x) = 5
Замена y = 2^(x) > 0 при любом x
y + 4/y = 5
y^2 - 5y + 4 = 0
D = (-5)^2 - 4*1*4 = 25 - 16 = 9 = 3^2
y1 = 2^(x) = (5 - 3)/2 = 2/2 = 1; x1 = 0
y2 = 2^(x) = (5 + 3)/2 = 8/2 = 4; x2 = 2

Ответ: x1 = 0; x2 = 2