How do you solve `sqrt(x^2-28)-1=x`?

Start by isolating the radical on one side of the equation.

`sqrt(x^2 - 28) - color(red)(cancel(color(black)(1))) + color(red)(cancel(color(black)(1))) = x+1`

Any possible solution to this equation must satisfy the conditions

• `x^2 - 28>=0` `->` for real numbers, you can only take the square root from positive numbers.
• `x-1>=0 => x>=1` `->` the square root of a real number can only be a positive number.

In order for the expression under the radical to be positive, you need

`x^2 - 28 >= 0`

`x^2 >= 28`

`|x| >=sqrt(28) => x<= -sqrt(28) vv x>= sqrt(28)`

But since `x>=1` is need for the right side of the equation, your overall condition will be `x>=sqrt(28)`.

So, square both sides of the equation to get rid of the radical

`(sqrt(x^2-28))^2 = (x+1)^2`

`color(red)(cancel(color(black)(x^2))) - 28 = color(red)(cancel(color(black)(x^2))) + 2x + 1`

This is equivalent to

`2x = -29 => x = color(red)(cancel(color(black)(-29/2)))`

As you can see, this is an extraneous solution because it doesn't meet the condition `x>=sqrt(28)`. This means that the original equation has no real solutions.