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

1 Answer
Stefan V.
Aug 19, 2015

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())

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