Solve by Factoring x^3+64=0

Problem

x^3+64=0

Solution

1. Identify the expression as a sum of cubes, noting that 64 can be written as 4^3

2. Apply the formula for the sum 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 by subtracting 4 from both sides.

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 discriminant and the resulting complex roots.

x=(4±√(,16−64))/2

x=(4±√(,−48))/2

x=(4±4*i√(,3))/2

x=2±2*i√(,3)

Final Answer

x=−4,2+2*i√(,3),2−2*i√(,3)

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