you are asked to nd all real and complex solutions of the gi

you are asked to nd all (real and complex) solutions of the given equation.

x^6 + 7x^3 - 8 = 0

Let x^3 =y

y^2 +7y -8 =0

y^2 +8y -y -8 =0

y(y +8) -1(y+8) =0

(y -1)(y +8) =0

y =1

x^3 =1 ----> x^3 -1 =0

(x-1)(x^2 +x +1) =0

x = 1 ; x = (1 +/- sqrt(1-4) )/2 = 0.5 +/- isqrt3

y +8 =0 ---> x^3 +8 =0

x = -2 , 1 +isqrt3 , 1 - isqrt3

Solution : x = -2 , 1 , 0.5 - isqrt3 , 0.5 + isqrt3, 1 +isqrt3 , 1 - isqrt3