Find all solutions to the following equation. $x ^ { 3 } - 8 = 0$
A) $x = 1,1 + i \sqrt { 3 } , 1 - i \sqrt { 3 }$
B) $x = 2 , - 2$
C) $x = - 2,2 - i \sqrt { 3 } , 2 + i \sqrt { 3 }$
D) $x = 2$
E) $x = 2 , - 1 + i \sqrt { 3 } , - 1 - i \sqrt { 3 }$

Question

Find all solutions to the following equation. $x ^ { 3 } - 8 = 0$ ​
A) $x = 1,1 + i \sqrt { 3 } , 1 - i \sqrt { 3 }$
B) $x = 2 , - 2$
C) $x = - 2,2 - i \sqrt { 3 } , 2 + i \sqrt { 3 }$
D) $x = 2$
E) $x = 2 , - 1 + i \sqrt { 3 } , - 1 - i \sqrt { 3 }$

Quizplus · Accepted Answer

The equation $x^3 - 8 = 0$ can be solved by adding 8 to both sides and then taking the cube root of both sides, resulting in $x^3 = 8$. The real solution is $x = 2$, as $2^3 = 8$. However, this is a cubic equation, which means it has three roots in total, including complex roots. The other two roots can be found using the fact that the sum of the roots of a cubic equation with real coefficients must be real. The complex roots are the cube roots of unity multiplied by the real root, leading to $x = -1 + i\sqrt{3}$ and $x = -1 - i\sqrt{3}$, which are the two complex conjugate roots.

Find All Solutions to the Following Equation x3−8=0x ^ { 3 } - 8 = 0x3−8=0

Find All Solutions to the Following Equation $x ^ { 3 } - 8 = 0$