Solve for x x^3-64=0

Problem

x^3−64=0

Solution

1. Identify the equation as a difference of cubes, noting that 64=4^3

2. Apply the formula for the difference of cubes, a^3−b^3=(a−b)*(a^2+a*b+b^2) where a=x and b=4

x^3−4^3=(x−4)*(x^2+4*x+16)

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

(x−4)*(x^2+4*x+16)=0

4. Solve the linear factor for the first root.

x−4=0⇒x=4

5. Solve the quadratic factor using the quadratic formula x=(−b±√(,b^2−4*a*c))/(2*a) for x^2+4*x+16=0

x=(−4±√(,4^2−4*(1)*(16)))/(2*(1))

6. Simplify the expression under the square root.

x=(−4±√(,16−64))/2

x=(−4±√(,−48))/2

7. Extract the imaginary unit and simplify the radical, noting √(,−48)=√(,−1⋅16⋅3)=4*i√(,3)

x=(−4±4*i√(,3))/2

8. Divide by the denominator to find the remaining complex roots.

x=−2±2*i√(,3)

Final Answer

x=4,−2+2*i√(,3),−2−2*i√(,3)

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