Question

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

Answer 1

`x = 2`

Explanation:

First, start by taking a look at the equation

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

Right from the start, you need `x` to be positive. Not only that, but you need the expression that's under the square root to be positive as well, since you cannot take the square root of a negative number and get a real number as a solution.

So, you know that you need

`x - 1 >=0 implies x>=1`

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

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

`x-1 = x^2/4`

This is equivalent to

`x^2 -4x + 4 = 0`

Use the quadratic formula to find the solutions to this quadratic equation

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

`x_(1,2) = (4 +- sqrt(0))/2`

This means that the quadratic has one distinct solution

`x = 4/2 = color(green)(2)`

Since `x =2` satisfies the condtion `x>=1`, this will also be the solution to your original equation.

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

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

`sqrt(1) = 1`

`1 = 1" "color(green)(sqrt())`