Question

How do you solve `n^2 + 19n + 66 = 6` by completing the square?

Answer 1

First simplify the process by moving the constant term to the right side of the equation;
then add whatever constant is necessary to both sides so that the left side is a square;
finally take the square root of both sides and simplify.

`n^2+19n+66 = 6`

`n^2+19n = -60`

If `n^2 + 19n` are the first two terms of a square
`(n+a)^2 = (n^2+2an+a^2)`
then
`a = 19/2`
and
`a^2 = (19/2)^2 = 361/4 = 90 1/4`

`n^2 + 19n + (19/2)^2 = -60 +90 1/4`

`(n+19/2)^2 = 30 1/4 = 121/4`

`n+19/2 = +-sqrt(121/4) = +- 11/2`

`n = -19/2 +- 11/2 = (-19 +-11)/2`

`n = -15`
or
`n=--4`

Answer 2

`n=-4, -15`

You can solve the problem by factoring.

`n^2+19n+66=6`

Subtract 6 from both sides.

`n^2+19n+60=0`

Factor.

`4xx15=60 and 4+15=19`

`(n+4)(n+15)=0`

`n+4=0`

`n=-4`

`n+5=0`

`n=-5`

`n=-4, -15`