How do you solve `x= sqrt(2x+3)`?

Right from the start, you know that you can only use positive values of `x`, since taking the square root of a real positive number will always produce a positive value.

This means that you need to have `x >=0`.

With this in mind, start by squaring both sides of the equation

`(sqrt(2x+3))^2 = x^2`

`2x + 3 = x^2`

Rearrange this equation by moving all the terms on one side

`x^2 - 2x - 3 = 0`

You can find the solutions to this quadratic equation by using the quadratic formula, which for a general form quadratic

`color(blue)(ax^2 + bx + c = 0)`

allows you to find the roots by using the formula

`color(blue)(x_(1,2) = (-b +- sqrt(b^2 - 4ac))/(2a))`

In your case, you can write

`x_(1,2) = (-(-2) +- sqrt((-2)^2 - 4 * 1 * (-3)))/(2 * 1)`

`x_(1,2) = (2 +- sqrt(16))/2`

`x_(1,2) = (2 +- 4)/2 = {(x_1 = (2 + 4)/2 = 3), (x_2 = (2 - 4)/2 = -1) :}`

SInce `x_2 = -1` does not satisfy the condition `x>=0`, your original equation will have only one solution, `x = color(green)(3)`.

You can do a quick check to make sure that the calculations are correct

`sqrt(2 * 3 + 3) = 3`

`sqrt(9) = 3`

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