Completing the Square with an Odd Middle Term

Okay, so in this video, we're going to complete the square again. The one that the reason this is a little bit harder is because your middle term, your 7 N, that's odd. So when you take half of it, you're going to end up getting a fraction. So we're going to start the same way. We're going to add the constant to the right hand side. And so we end up with N squared plus 7 N plus something is equal to negative three plus something. This time though, if we take half in the middle term, we get 7 halves, and if we square that, we get 49, fourths. So we're going to add 49 fourths to both sides. So the left hand side is still N squared plus 7 N plus 49 fourths. And the right hand side is going to be negative three plus 49 fourths. So that's the same as negative 12 fourths plus 49 fourths. Which gives us 37 fourths. The next step is going to be to factor the left hand side, so we're going to get N plus and the reason I got plus is because the middle term is positive. Again, same two factors because we have a perfect square trinomial. The easiest way to get this part of the factor this last term is it's always just going to be half in the middle term. Or if you want to think of it this way, it's just the square root of the third term. And then the right hand side is going to be 37 fourths. So this is really just N plus 7 halves squared is equal to 37 fourths. Take the square root of both sides. And we end up with N plus 7 over two is equal to and I would probably split this up. So plus or minus the square root of 37 over the square root of four. And we get N plus 7 halves is equal to plus or minus the square root of 37, which is irrational, and there's no perfect square factor. So you just leave it. And then the square root of four is two. And then your last step is to subtract 7 halves from both sides. So your final answer is N equals negative 7 plus or minus the square root of 37. All over two.