Question

How would you simplify `sqrt(2x+3) - sqrt(x+1 )=1`?

Answer 1

You must mean how to solve. I'll solve the equation for you.

Explanation:

`sqrt(2x + 3) - sqrt(x + 1) = 1`

`sqrt(2x + 3) = 1 + sqrt(x + 1)`

`(sqrt(2x + 3))^2 = (1 + sqrt(x + 1))^2`

`2x + 3 = 1 + 2sqrt(x + 1) + x + 1`

`2x - x + 3 - 1 - 1 = 2sqrt(x + 1)`

`x + 1 = 2sqrt(x + 1)`

`(x + 1)^2 = (2sqrt(x + 1))^2`

`x^2 + 2x + 1 = 4(x + 1)`

`x^2 + 2x + 1 = 4x + 4`

`x^2 + 2x - 4x + 1 - 4 = 0`

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

`(x - 3)(x + 1) = 0`

`x = 3 and x = -1`

Always check the solutions in the original equation to make sure they aren't extraneous. If they do not work in the original equation, you must reject them.

`sqrt(2 xx 3 + 3) - sqrt(3 + 1) = 1`

So, x = 3 works. Now, let's check x = -1:

`sqrt(2 xx -1 + 3) - sqrt(-1 + 1) = 1`

So, x = -1 works as well.

Your solution set would be x = 3, -1

Practice exercises:

1. Solve for x.

a) `sqrt(3x - 2) - sqrt(x - 2) = 2`

b) `sqrt(4x + 5) + sqrt(8x + 9) = 12`