Question

How do you solve `x+5=sqrt(17+x)?`?

Answer 1

`x=-1`

Explanation:

`x+5=sqrt(17+x)`

Square both sides
`x^2+10x+25=17+x`

Subtract to make one side 0
`x^2+9x+8=0`

Factor
`(x+8)(x+1)=0`

Solve for x:
`x+8=0 or x+1=0`
`x=-8 or x=-1`

Since we squared both sides, we must check for extraneous solutions:
`-1+5=sqrt(17-1)`
`4=4`, `x=-1`

`-8+5=sqrt(17-8)`
`-3=sqrt(9)`
By convention, the `sqrt()` symbol always refers to the positive square root, so:
`-3!=3`, `x!=-8`

Therefore, `x=-1`

Answer 2

`x = -1`

Explanation:

`x + 5 = sqrt(17+x)`

First, square both sides:
`(x+5)^color(blue)2 = (sqrt(17+x))^color(blue)2`

`(x+5)^2 = 17+x`

We know that any binomial in the form of:
`(x+y)^2 = x^2 + 2xy + y^2`, so the left hand side becomes:
`x^2 + 10x + 25`

Put that back into the equation:
`x^2 + 10x + 25 = 17 + x`

Subtract `color(blue)(17)` and `color(blue)x` from both sides:
`x^2 + 9x + 8 = 0`

This is in standard form, or `color(red)(a)x^2 + color(green)(b)x + color(magenta)(c)`. In this form, we can factor it with two numbers that:

1. Multiply up to `color(red)(a)* color(magenta)(c) = color(red)(1) * color(magenta)(8) = 8`
2. Add up to `color(green)(b)`, or `color(green)9`

Those two numbers are `color(blue)8` and `color(blue)1`, as:

1. `1 * 8 = 8`
2. `1 + 8 = 9`

Therefore, the factored form is:
`(x + 8)(x + 1) = 0`

Since when the expressions multiply to equal zero, that means each of the expressions equal zero. So we can set up two equations:
`x+8 = 0` and `x+1 = 0`

Subtract `color(blue)8` in the first equation, subtract `color(blue)1` in the second equation:
`x = -8` and `x = -1`

`--------------------`

Now we have to check if both solutions are really solutions by plugging them into the original formula. Let's check `color(blue)(-8)` first:

`x + 5 = sqrt(17+x)`

`-8 + 5 = sqrt(17-8)`

`-3 = sqrt9`

`-3 != 3`

No, this is not true, so `color(blue)(-8)` is NOT a solution! Now check `color(blue)(-1)`:
`-1 + 5 = sqrt(17-1)`

`4 = sqrt16`

`4 = 4`

Yes, this is true, so `color(blue)(-1)` is the ONLY solution!

Hope this helps!