task/30362718 (4ˣ -8)² - 10|4ˣ -8| =3*4ˣ - 36
решение замена : t = 4ˣ
(t -8)² - 10|t -8| =3t - 36
а) t < 8
(t - 8)²+10(t - 8) =3t - 36 ⇔t²- 16t + 64 + 10t - 80 - 3t + 36= 0 ⇔
t²-9t +20 =0 ⇒[ t=4 ; t=5. * * * 4; 5 ∈ (-∞ ,8) * * * [ 4ˣ = 4 ; 4ˣ =5 . ⇔
[ x =1 ; x =lg5 / lg4 .
б) t ≥ 8
(t - 8)²- 10(t - 8) = 3t - 36 ⇔ t² - 16t + 64 - 10t + 80 - 3t +36 = 0 ⇔
t²-29t +180 =0 || D =29² -4*180 =(30-1)² -720 =841 -720 =121 =11² ||
t₁,₂= (29 ±11)/2 . [ t =9 ; t =20 . * * * 9 ; 20 ∈ [ 8 ;∞) * * *
[4ˣ =9 ; 4ˣ =20 . ⇔
[ x =lg9 / lg4 = lg3 / lg2 ; x =lg20 / lg4 = (1+lg2)/2lg2 .
ответ : 1 ; lg5 / lg4 ; lg9 / lg4 ; lg20 / lg4 .
* * * 4ˣ =4 ; 4ˣ =5 ; 4ˣ =9 ; 4ˣ =20 * * *