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Find-the-value-of-x-x-3-64-0-Mastermind-




Question Number 168527 by Mastermind last updated on 12/Apr/22
Find the value of x  x^3 +64=0    Mastermind
$${Find}\:{the}\:{value}\:{of}\:{x} \\ $$$${x}^{\mathrm{3}} +\mathrm{64}=\mathrm{0} \\ $$$$ \\ $$$${Mastermind} \\ $$
Answered by Mathspace last updated on 12/Apr/22
x^3 +64=0⇔x^3 +4^3 =0 ⇔  (x+4)(x^2 −4x+16)=0 ⇔x=−4  or x^2 −4x+16=0  Δ^′ =(−2)^2 −16=4−16=−12 ⇒  the?roots are z_1 =2+2i(√3)  z_2 =2−2i(√3)  the solution of this equation is  −4,z_1 and z_2
$${x}^{\mathrm{3}} +\mathrm{64}=\mathrm{0}\Leftrightarrow{x}^{\mathrm{3}} +\mathrm{4}^{\mathrm{3}} =\mathrm{0}\:\Leftrightarrow \\ $$$$\left({x}+\mathrm{4}\right)\left({x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{16}\right)=\mathrm{0}\:\Leftrightarrow{x}=−\mathrm{4} \\ $$$${or}\:{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{16}=\mathrm{0} \\ $$$$\Delta^{'} =\left(−\mathrm{2}\right)^{\mathrm{2}} −\mathrm{16}=\mathrm{4}−\mathrm{16}=−\mathrm{12}\:\Rightarrow \\ $$$${the}?{roots}\:{are}\:{z}_{\mathrm{1}} =\mathrm{2}+\mathrm{2}{i}\sqrt{\mathrm{3}} \\ $$$${z}_{\mathrm{2}} =\mathrm{2}−\mathrm{2}{i}\sqrt{\mathrm{3}} \\ $$$${the}\:{solution}\:{of}\:{this}\:{equation}\:{is} \\ $$$$−\mathrm{4},{z}_{\mathrm{1}} {and}\:{z}_{\mathrm{2}} \\ $$

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