Question

How do you solve `sqrt(x^2+2)-5=0`?

Answer 1

`x=sqrt23, x=-sqrt23`

Explanation:

`sqrt(x^2+2) - 5 = 0`

Add 5 to each side:
`sqrt(x^2+2) = 5`

Square each side:
`x^2+2 =25`

Subtract 2 from each side:
`x^2 = 23`

`color(blue)(x= +- sqrt23)`

Answer 2

See a solution process below:

Explanation:

First, add `color(red)(5)` to each side of the equation to isolate the radical while keeping the equation balanced:

`sqrt(x^2 + 2) - 5 + color(red)(5) = 0 + color(red)(5)`

`sqrt(x^2 + 2) - 0 = 5`

`sqrt(x^2 + 2) = 5`

Next, square each side of the equation to eliminate the radical while keeping the equation balanced:

`(sqrt(x^2 + 2))^2 = 5^2`

`x^2 + 2 = 25`

Then, subtract `color(red)(2)` from each side of the equation to isolate `x^2` while keeping the equation balanced:

`x^2 + 2 - color(red)(2) = 25 - color(red)(2)`

`x^2 + 0 = 23`

`x^2 = 23`

Now, take the square root of each side of the equation to solve for `x` while keeping the equation balanced. Remember, the square root of a number produces both a positive and a negative result:

`sqrt(x^2) = +-sqrt(23)`

`x = +-sqrt(23)`