Solve by Factoring x^3+8=0

Problem

x^3+8=0

Solution

1. Identify the expression on the left side as a sum of two cubes, since x^3 is a cube and 8=2^3

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

3. Factor the equation into a linear factor and a quadratic factor.

(x+2)*(x^2−2*x+4)=0

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

x+2=0

x^2−2*x+4=0

5. Solve the linear equation for the first root.

x=−2

6. Apply the quadratic formula to solve x^2−2*x+4=0 where a=1 b=−2 and c=4

x=(−(−2)±√(,(−2)^2−4*(1)*(4)))/(2*(1))

7. Simplify the discriminant and the resulting complex roots.

x=(2±√(,4−16))/2

x=(2±√(,−12))/2

x=(2±2*i√(,3))/2

x=1±i√(,3)

Final Answer

x=−2,1+i√(,3),1−i√(,3)

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