Solve for x x^3-216=0

Problem

x3−216=0

Solution

1. Identify the equation 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

x3−6=(x−6)*(x2+6*x+36)

3. Set each factor to zero using the zero product property.

x−6=0

x2+6*x+36=0

4. Solve the linear equation for the first root.

x=6

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

x=(−6±√(,6−4*(1)*(36)))/(2*(1))

6. Simplify the discriminant and the resulting expression.

x=(−6±√(,36−144))/2

x=(−6±√(,−108))/2

7. Extract the imaginary unit and simplify the radical √(,108)=√(,36⋅3)=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)

Want more problems? Check here!