Question

How do you solve and check for extraneous solutions in `x^2 + 2 = x + 4`?

Answer 1

Express the given equation in standard form, factor for solution values, and check each resulting value to ensure that it is not extraneous.
`x=2` or `x=-1`

Explanation:

Given `x^2+2 = x+4`

Subtract `(x+4)` from both sides to get into standard form
`color(white)("XXXX")` `x^2-x-2=0`

Factor
`color(white)("XXXX")` `(x-2)(x+1) = 0`

Either `(x-2)=0` or `(x+1)=0`
`rArr` `color(white)("XXXX")` `x=2` or `x=-1`

Checking `x=2` by substituting into `x^2+2` and `x+4`
`color(white)("XXXX")` `x^2+2=(2)^2+2 = 6`
`color(white)("XXXX")` `x+4=(2)+4 = 6`
`x^2+2=x+4` so `x=2` is not extraneous.

Similarly, checking `x=-1` by substituting into `x^2+2` and `x+4`
`color(white)("XXXX")` `x^2+2=(-1)^2+2 = 3`
`color(white)("XXXX")` `x+4=(-1)+4 = 3`
`x^2+2=x+4` so `x=-1` is not extraneous.