How to solve square root equations?

Example: solve √ (2x-5) - √ (x-1) = 1 isolate one of the square roots: √ (2x-5) = 1 + √ (x-1) square both sides: 2x-5 = (1 + √ (x-1)) 2

We have removed one square root.

Expand right hand side: 2x-5 = 1 + 2√ (x-1) + (x-1)

Simplify: 2x-5 = 2√ (x-1) + x

Subtract x from both sides: x-5 = 2√ (x-1)

Now do the "square root" thing again:

isolate the square root: √ (x-1) = (x-5) / 2 square both sides: x-1 = ((x-5) / 2) 2

We have now successfully removed both square roots.

Let us continue on with the solution.

Expand right hand side: x-1 = (x2 - 10x + 25) / 4

It is a Quadratic Equation! So let us put it in standard form.

Multiply by 4 to remove division: 4x-4 = x2 - 10x + 25

Bring all to left: 4x - 4 - x2 + 10x - 25 = 0

Combine like terms: - x2 + 14x - 29 = 0

Swap all signs: x2 - 14x + 29 = 0

Using the Quadratic Formula (a=1, b=-14, c=29) gives the solutions:

2.53 and 11.47 (to 2 decimal places)

Let us check the solutions:

2.53: √ (2·2.53-5) - √ (2.53-1) ≈ - 1 Oops! Should be plus 1!

11.47: √ (2·11.47-5) - √ (11.47-1) ≈ 1 Yes that one works.

There is really only one solution:

Answer: 11.47 (to 2 decimal places)