Question

Find all real solutions and show work:
2x^3-3x^2+18x-27=0

Answer 1

Given:

The given equation is

`2x^3-3x^2+18x-27=0`

To find:

All the real solutions.

Solution:

We have,

`2x^3-3x^2+18x-27=0`

It can be written as

`x^2(2x-3)+9(2x-3)=0`

`(2x-3)(x^2+9)=0`

Using zero product property, we get

`(2x-3)=0` and
`(x^2+9)=0`

`2x=3` and
`x^2=-9`

`x=(3)/(2)` and
`x=\pm√(-9)`

We know that
`x=\pm√(-9)` is an imaginary number because there is a negative sign under the square root.

Therefore,
`x=(3)/(2)` is the only real solution of the given equation.