You certainly got the right answer, but that's so many steps, and so many chances to make a mistake. I feel like there's a much easier way: (m-4)³ = -8 => m-4 = ³√(-8) => m = 4 + ³√(-8) Then you just need to find the three cube roots of -8. Which ought to be much easier to solve
(Personally, I used a geometric argument to find the cube roots of -8. I could see that -2 is one of the three roots, and I knew that the other two would have the same magnitude, and be equally spaced around the origin on the complex plane. If you know the right trigonometry facts, you can jump straight to (1+i√3) and (1-i√3) )
Could also use your cubic formula to find the cube root of -8 If we use x³ + 8 = 0, we get: (x+2)(x²-2x+4)=0. Therefore either x=-2, or: (x²-2x+4) = 0 (x-1)²-1+4 = 0 (x-1)² = -3 x-1 = √(-3) x = 1 ± i√3 …okay, once it's all written out I admit that it isn't any shorter than your solution. But I still think it's easier to work with the smaller numbers
(m-4)^3=-8 m=2 m=5 ± Sqrt[3]i
Very nice❤️❤️
You certainly got the right answer, but that's so many steps, and so many chances to make a mistake. I feel like there's a much easier way:
(m-4)³ = -8
=> m-4 = ³√(-8)
=> m = 4 + ³√(-8)
Then you just need to find the three cube roots of -8. Which ought to be much easier to solve
(Personally, I used a geometric argument to find the cube roots of -8. I could see that -2 is one of the three roots, and I knew that the other two would have the same magnitude, and be equally spaced around the origin on the complex plane. If you know the right trigonometry facts, you can jump straight to (1+i√3) and (1-i√3) )
Could also use your cubic formula to find the cube root of -8
If we use x³ + 8 = 0, we get:
(x+2)(x²-2x+4)=0. Therefore either x=-2, or:
(x²-2x+4) = 0
(x-1)²-1+4 = 0
(x-1)² = -3
x-1 = √(-3)
x = 1 ± i√3
…okay, once it's all written out I admit that it isn't any shorter than your solution. But I still think it's easier to work with the smaller numbers
You are a genius and so sweet my friend ❤️❤️
@@MathOlympiad0 why thank you!
we get , m^3-12m^2+48m-56=0 , by fakt. , --> , (m-2 )(m^2-10m+28)=0 , m= 4 , 5+V3 , 5-V3 ,
You are a genius and so sweet my friend ❤️❤️
@@MathOlympiad0 Thanks , I am constantly learning from you about the great data solutions.