What is the coordinates of the point of intersection of the curve `y=1/(x+2) +3` and the line `y=-x`?

To find an intersection we try to find an `x`-value where both equations give us the same `y` value.

So we want to solve:

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

We'll begin by clearing the fraction. Multiply both sides of the equation by `(x+2)`

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

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

`1+3x+6 = -x^2-2x`

We see an `x^2`, so this is a qudratic equation. The usual way of writing a quadratic equation is to have all of the terms on one side equal to `0` on the other, so let's make it like that.

`x^2+5x+7 = 0`.

We try to solve by factoring, but we don't see how to factor this, so either solve by completing the square or solve using the quadratic formula.

`x = (-b+-sqrt(b^2-4ac))/(2a)`

`x = (-(5)+-sqrt((5)^2-4(1)(7)))/(2(1))`

But now we notice (if we didn't notice before) that

`sqrt(b^2-4ac) = sqrt((5)^2-4(1)(7)) = sqrt(25-28) = sqrt(-3)`.

So there is no solution in the real numbers.

Therefore, there is no point of intersection for the curves.

Here is the graph. Of course, `y=-x` is the straight line.

graph{(y+x)(y-1/(x+2)-3)=0 [-17.02, 15.01, -5.2, 10.82]}