Solve by Factoring x^3-216=0

Problem

x3−216=0

Solution

1. Identify the expression as a difference of cubes, noting that 216 is 6

2. Apply the formula for the difference of cubes, a3−b3=(a−b)*(a2+a*b+b2) where a=x and b=6

(x−6)*(x2+6*x+36)=0

3. Set the first factor to zero to find the real solution.

x−6=0

x=6

4. Set the second factor to zero and apply the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a) to find the remaining solutions.

x2+6*x+36=0

5. Substitute the values a=1 b=6 and c=36 into the quadratic formula.

x=(−6±√(,6−4*(1)*(36)))/(2*(1))

6. Simplify the expression under the radical.

x=(−6±√(,36−144))/2

x=(−6±√(,−108))/2

7. Simplify the radical using the imaginary unit i and the fact that √(,108)=6√(,3)

x=(−6±6*i√(,3))/2

x=−3±3*i√(,3)

Final Answer

x=6,−3+3*i√(,3),−3−3*i√(,3)

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