Question

Determine for which values of x, √(2x^(2)-1)/x=x will have no roots?

Answer 1

`x= +-1`

Explanation:

`sqrt(2x^(2)-1)/x=x`
`sqrt(2x^(2)-1)= x^2`
`2x^(2)-1 = (x^2)^2`

Let x^2 = m
`2m - 1 = m^2`
`m^2 - 2m + 1 = 0`
`m = +-1`

`x^2 = m`
`x^2 = -1 =>` no real solutions
`x^2 = 1 => x= +-1` and both values works in the original equation:

Check:
`sqrt(2xx1^(2)-1)/1=1`
`sqrt(2xx(-1)^(2)-1)/-1=-1`