Solve by Factoring x^3+27=0

Problem

x3+27=0

Solution

1. Identify the expression on the left side as a sum of two cubes, since x3 is a cube and 27=3

2. Apply the sum of cubes formula, which is a3+b3=(a+b)*(a2−a*b+b2) where a=x and b=3

(x+3)*(x2−3*x+9)=0

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

x+3=0

x2−3*x+9=0

4. Solve the linear equation for the first root.

x=−3

5. Solve the quadratic equation using the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a) where a=1 b=−3 and c=9

x=(−(−3)±√(,(−3)2−4*(1)*(9)))/(2*(1))

x=(3±√(,9−36))/2

x=(3±√(,−27))/2

6. Simplify the radical using the imaginary unit i noting that √(,−27)=√(,−1⋅9⋅3)=3*i√(,3)

x=(3±3*i√(,3))/2

Final Answer

x=−3,(3±3*i√(,3))/2

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