Solve by Factoring x^3-1=0

Problem

x3−1=0

Solution

1. Identify the expression on the left side as a difference of cubes, which follows the general form a3−b3=(a−b)*(a2+a*b+b2)

2. Apply the formula for the difference of cubes where a=x and b=1

x3−1=(x−1)*(x2+x+1)

3. Set each factor to zero according to the zero product property to find the roots of the equation.

(x−1)*(x2+x+1)=0

4. Solve the linear factor for the first root.

x−1=0⇒x=1

5. Solve the quadratic factor using the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a) where a=1 b=1 and c=1

x=(−1±√(,1−4*(1)*(1)))/(2*(1))

6. Simplify the expression under the radical to find the complex roots.

x=(−1±√(,−3))/2

x=(−1±i√(,3))/2

Final Answer

x=1,(−1+i√(,3))/2,(−1−i√(,3))/2

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