how do i find the real or imaginary number to 125x^3-27=0

125x^3-27 = 0

Factor the left side as a difference of two cubes:

a^3 - b^3 = (a-b)(a^2 +ab + b^2)

In the problem:

a = 5x and b = 3, since (5x)^3 = 125x^3 and 3^3 = 27

(5x-3)(25x^2 + 15x + 9) = 0

As William pointed out, setting 5x-3 = 0 gives you x = 3/5.

To get any other possible roots, you can do the following:

25x^2 + 15x + 9 = 0

-15 +- sqrt(225 - 4(25)(9))

x = ---------------------------------

50

x = -15 +- sqrt(225 - 900)

--------------------

50

x = -15 +- sqrt(-675)

-------------------

50

x = -15 +- sqrt( (-1)(25)(9)(3) )

--------------------------------

50

x = -15 +- 15i sqrt(3)

-------------------

50

x = -3 +- 3i √3

------------

10

Don't forget the 3/5:

x = {3/5, -3/10 + (3i√3)/10, -3/10 - (3i√3)/10}

Factor the left side as a difference of two cubes:

a^3 - b^3 = (a-b)(a^2 +ab + b^2)

In the problem:

a = 5x and b = 3, since (5x)^3 = 125x^3 and 3^3 = 27

(5x-3)(25x^2 + 15x + 9) = 0

As William pointed out, setting 5x-3 = 0 gives you x = 3/5.

To get any other possible roots, you can do the following:

25x^2 + 15x + 9 = 0

-15 +- sqrt(225 - 4(25)(9))

x = ---------------------------------

50

x = -15 +- sqrt(225 - 900)

--------------------

50

x = -15 +- sqrt(-675)

-------------------

50

x = -15 +- sqrt( (-1)(25)(9)(3) )

--------------------------------

50

x = -15 +- 15i sqrt(3)

-------------------

50

x = -3 +- 3i √3

------------

10

Don't forget the 3/5:

x = {3/5, -3/10 + (3i√3)/10, -3/10 - (3i√3)/10}