Solve by Factoring 8x^3-27=0

Problem

8*x3−27=0

Solution

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

2. Determine the values of a and b by taking the cube roots of the terms: a=√(3,8*x3)=2*x and b=√(3,27)=3

3. Apply the formula for the difference of cubes to factor the equation.

(2*x−3)*((2*x)2+(2*x)*(3)+3)=0

4. Simplify the terms inside the second factor.

(2*x−3)*(4*x2+6*x+9)=0

5. Set each factor to zero to find the solutions.

2*x−3=0

4*x2+6*x+9=0

6. Solve the linear equation for the real root.

2*x=3

x=3/2

7. Solve the quadratic equation using the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a) to find the complex roots.

x=(−6±√(,6−4*(4)*(9)))/(2*(4))

8. Simplify the discriminant and the resulting roots.

x=(−6±√(,36−144))/8

x=(−6±√(,−108))/8

x=(−6±6*i√(,3))/8

x=(−3±3*i√(,3))/4

Final Answer

x=3/2,(−3±3*i√(,3))/4

Want more problems? Check here!