Question

How do you solve the equation `x^2+2x+6=0` by completing the square?

Answer 1

There are no Real solutions to this equation
but see below for method of completing the square
and for Complex value solutions.

Explanation:

Remember that a squared binomial takes the form
`color(white)("XXX")(x=color(red)a)^2=x^2+2color(red)ax+color(red)a^2`

So if `x^2+2x` are the first 2 terms of an expanded squared binomial:
`color(white)("XXX")color(red)a=1`
and the third term must be `color(red)(a)^2=1^2=1`
and the complete (expanded) square needs to be `color(blue)(x^2+2x+1)`

Rewriting the given equation: `x^2+2x+6=0`
to include this complete (expanded) square:
`color(white)("XXX")color(blue)(x^2+2x+1)+5=0`

or
`color(white)("XXX")color(blue)(x^2+2x+1) = -5`

or
`color(white)("XXX")color(blue)(""(x+1)^2)=-5`

We could note at this point that since any Real number squared is greater than or equal to `0` there can be no Real solutions.

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If you have advanced to Complex numbers:
`color(white)("XXX")color(blue)(x+1)=+-sqrt(-5) =+-isqrt(5)`
and
`color(white)("XXX")x=-1+-isqrt(5)`