Solve by Factoring x^3-64=0

Problem

x3−64=0

Solution

1. Identify the expression on the left side as a difference of cubes, which follows the pattern a3−b3=(a−b)*(a2+a*b+b2)

2. Rewrite the constant term as a cube to find the values of a and b

x3−4=0

3. Apply the formula for the difference of cubes where a=x and b=4

(x−4)*(x2+4*x+16)=0

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

x−4=0

x2+4*x+16=0

5. Solve the linear equation for the first root.

x=4

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

x=(−4±√(,4−4*(1)*(16)))/(2*(1))

7. 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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