Question

How do you solve `sqrt(x + 2)+4 = x`?

Answer 1

`x = 7`

Explanation:

Start by isolating the radical on one side of the equation by adding `-4` to both sides

`sqrt(x+2) + color(red)(cancel(color(black)(4))) - color(red)(cancel(color(black)(4))) = x-4`

`sqrt(x+2) = x-4`

This means that, in order to be considered a valid solution, the value(s) of `x` must satisfy the condition

`x-4>=0 <=> x>=4`

Next, square both sides of the equation to get rid of the square root

`(sqrt(x+2))^2 = (x-4)^2`

`x+2 = x^2 - 8x + 16`

Rearrange this equation to get all the terms on one side

`x^2 - 9x + 14 = 0`

You can find the two solutions to this quadratic by using the quadratic formula

`x_(1,2) = (-(-9) +- sqrt( (-9)^2 - 4 * 1 * (-14)))/(2 * 1)`

`x_(1,2) = (9 + sqrt(25))/2 = (9 +-5)/2 = {(x_1 = (9+5)/2 = color(green)(7)), (x_2 = (9-5)/2 = cancel(color(red)(2))) :}`

The only valid solution will be `x=7`; `x=2` will be an extraneous solution, since it does not satisfy the aforementioned condition `x>=4`.