Question

How do you solve `sqrt(x+6)=x`?

Answer 1

`x=3`

Explanation:

Start by squaring both sides of the equation to get rid of the square root

`(sqrt(x+6))^2 = x^2`

`x + 6 = x^2`

Next, move all the tems on one side of the equation

`x^2 - x - 6 = 0`

Use the quadratic formula to find the two roots of this quadratic equation

`x_(1, 2) = (-(-1) +- sqrt( (-1)^2 - 4 * 1 * (-6)))/(2 * 1)`

`x_(1,2) = (1 +- sqrt(25))/2 = (1 +- 5)/2 = { (x_1 = (1 + 5)/2 = 3), (x_2 = (1 - 5)/2 = -2) :}`

Notice that `x=-2` does not satisfy the initial equation, since you would get

`sqrt(-2 + 6) = -2`

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

This means that `x=-2` will eb an extraneous solution. The only valid solution will thus be `x = 3`, for which

`sqrt(3 + 6) = 3`

`sqrt(9) = 3 " "color(green)(sqrt())`