Solve the equation `x(x+2)(x^2-1)=-1.`

Answer

`x= (-1pmsqrt(5))/(2)`

Explanation

We have
`x(x + 2) (x^(2) - 1) = -1`
or `x(x + 2)(x + 1) (x - 1) = -1`
or `(x(x + 1))(x + 2) (x - 1) = -1`
`(x^(2) + x)(x^(2) + x - 2) = -1`
`(x^(2) + x)^(2)-2x(x^(2) + x) +1 = 0`
Putting `x^(2) x = y, we get y^(2) - 2y+ 1 = 0`
or `(y - 1)^(2) = 0`
or `y=1`
Now `x^(2) + x = 1`
or `x^(2) + x -1 = 0`
or `x= (1-pmsqrt(1+4))/(2)=(-1pmsqrt(5))/(2)`
Hence, the roots of the given equation are
`(-1+sqrt(5))/(2),(-1+sqrt(5))/(2),(-1-sqrt(5))/(2), and (-1-sqrt(5))/(2)` .