Solve by Factoring x^3-8=0

Problem

x3−8=0

Solution

1. Identify the expression on the left side as a difference of cubes, noting that 8=2

2. Apply the formula for the difference of cubes, a3−b3=(a−b)*(a2+a*b+b2) where a=x and b=2

x3−2=(x−2)*(x2+2*x+4)

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

(x−2)*(x2+2*x+4)=0

4. Solve the linear factor by setting x−2=0

x=2

5. Solve the quadratic factor by setting x2+2*x+4=0 and applying the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a)

x=(−2±√(,2−4*(1)*(4)))/(2*(1))

6. Simplify the result under the radical to find the complex solutions.

x=(−2±√(,−12))/2

x=(−2±2*i√(,3))/2

x=−1±i√(,3)

Final Answer

x=2,−1+i√(,3),−1−i√(,3)

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