What number must you add to complete the square? x^2+6x=19

The number 100 is added to complete the square

Well, the idea being, how do we turn x²+6x=19 with a perfect square trinomial

well, recall, the middle term of a perfect square trinomial is just 2 * guy on the left * guy on the right

so ... with that in mind, let's nevermind the 19, and take close peek at the 6x, let's do some grouping

(x² + 6x + ?) = 19

well, 6x is really 2 * 3 * x

since the middle term is just 2 * guy on one side * guy on the other

we can tell one guy is x², already there, but the other will then be 3²

now, let's go to our Mr very good friend Mr Zero, 0, and add that 3², however, if we add whatever, we must also subtract whatever, we're only borrowing from zero

(x² + 6x + 3²) - 3² = 19

(x + 3) ² - 9 = 19

(x + 3) ² = 28

so, the number we needed was just 3² or 9, courtesy of zero and our fellow the middle term