How to find the roots of x3 2 0 I am not allowed to use a

How to find the roots of: x³ - 2 = 0

I am not allowed to use a calculator on my exam thus graphing is of no use to me. If anyone could show me the strategy with a step by step for how to solve this by hand, that would be very helpful. I know one of the ovbious roots is 3rd root of sq root(2). I used an online calculator to see the other two roots are imaginary, but how can I figure this out by hand? Thank you.

Solution

x^3 - 2 =0

we can write this : x^3 - (2^1/3)^3 =0

We can solve this equation by factorisation and then equating each factor equal to 0

Use the formula : a^3 - b^3 =  (a-b)(a^2+ab+b^2).

x^3 - (2^1/3)^3 = (x- 2^1/3)(x^2 +x.2^1/3 + 2^2/3 )

(x- 2^1/3)(x^2 +x.2^1/3 + 2^2/3 ) =0

So, equate the factors to zero

x-2^1/3 =0 ----> x= 2^1/3

x= 1.259

(x^2 +x.2^1/3 + 2^2/3 ) =0

solve the quadratic using quadratice root formula:

x = (-b +/- sqrt(b^2 -4ac)/2a

x = (-2^1/3 + /- sqrt( 2^2/3 - 2^8/3) )/2

= -2^-2/3 + /- sqrt(2^2/3 -2^8/3)/2

= -0.63 +/- sqrt(-4.76)/2

= -0.63 +/- i*1.09

= -0.63 + i*1.09 and -0.63 - i*1.09

Roots are x = 1.259, -0.63 + i*1.09 and -0.63 - i*1.09