Solve by Factoring x^3-27=0

Problem

x3−27=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

x3−3=0

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

(x−3)*(x2+3*x+9)=0

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

x−3=0

x2+3*x+9=0

5. Solve the linear equation for the first root.

x=3

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

x=(−3±√(,3−4*(1)*(9)))/(2*(1))

7. Simplify the discriminant and the resulting complex roots.

x=(−3±√(,9−36))/2

x=(−3±√(,−27))/2

x=(−3±3*i√(,3))/2

Final Answer

x=3,(−3±3*i√(,3))/2

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