Question-174253

Question Number 174253 by mathlove last updated on 28/Jul/22

Answered by mnjuly1970 last updated on 28/Jul/22

x^( 3) +3x^( 2) +3x +1 = 2023x (x+1)^( 3) = 2023x x+1= ((2023))^(1/3) .(x)^(1/3) a+b+c+3=((2023))^(1/3) ((a)^(1/3) +(b)^(1/3) +(c)^(1/3) ) ⇒ 0= (a)^(1/3) +(b)^(1/3) +(c)^(1/3) ■

$$\:\:\:{x}^{\:\mathrm{3}} +\mathrm{3}{x}^{\:\mathrm{2}} +\mathrm{3}{x}\:+\mathrm{1}\:=\:\mathrm{2023}{x} \\ $$$$\:\:\:\:\:\left({x}+\mathrm{1}\right)^{\:\mathrm{3}} =\:\mathrm{2023}{x} \\ $$$$\:\:\:\:\:\:\:{x}+\mathrm{1}=\:\sqrt[{\mathrm{3}}]{\mathrm{2023}}\:.\sqrt[{\mathrm{3}}]{{x}} \\ $$$$\:\:\:\:\:{a}+{b}+{c}+\mathrm{3}=\sqrt[{\mathrm{3}}]{\mathrm{2023}}\:\left(\sqrt[{\mathrm{3}}]{{a}}\:+\sqrt[{\mathrm{3}}]{{b}}\:+\sqrt[{\mathrm{3}}]{{c}}\:\right) \\ $$$$\:\:\Rightarrow\:\:\:\:\:\mathrm{0}=\:\sqrt[{\mathrm{3}}]{{a}}\:+\sqrt[{\mathrm{3}}]{{b}}\:+\sqrt[{\mathrm{3}}]{{c}}\:\:\:\:\:\:\blacksquare \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\: \\ $$

Commented by Rasheed.Sindhi last updated on 28/Jul/22