Решить уравнение :
∛ (x -3) - ∛ (x -10) =1 ;
( ∛ (x -3) - ∛ ( x -10) )³ =1³ ;
(x -3) - ( x -10) - 3∛ (x -3) ∛ (x -10)* ( ∛ (x -3) - ∛ (x -10) )=1 ;
7 - 3∛ ((x -3) (x -10)) *1 =1 ;
3∛ ((x -3) (x -10)) = 6 ;
∛ ((x -3) (x -10)) = 2 ;
(x -3) (x -10) =8 ;
x² -13x+22 =0 ; * * * x² -( 2+11)*x+2*11 =0 * * *
D =13² -4*1*22 =169 -88 =81 =9²
x₁= (13 -9)/2 =2 ;
x₂ =(13+9) /2 =11 .
Проверка : ∛ (2 -3) - ∛ (2 -10) =∛ -1 - ∛ -8 = - 1 -(-2) =-1+2 =1 и
∛ (11-3) - ∛ ( 11-10) = ∛ 8 - ∛ 1 =2 -1 =1.
ответ : 2 и 11 .
* * * * * * * * * * *
Использовали формулу: (a- b)³ = a³ - b³ -3a² b+3a b² -b³ = a³ - b³ -3ab(a -b) .