Factoring two-variable quadratics: rearranging

let's see if we can use our existing factoring skills to factor 30 x squared plus 11x y plus y squared I encourage you to pause the video and see if you can handle it yourself now the first hint I will give you and this might kind of open up what's going on here is to maybe rearrange this a little bit we could rewrite this as y squared plus 11x Y plus 30x squared and my whole motivation for doing that it there are ways to factor in a quadratic where your first coefficient your coefficient on this first term is something other than 1 but we haven't seen that yet and so rearranging it this way this kind of got is a little bit more into our comfort zone now our coefficient is a 1 on the y squared term so now we can start to think of our think think of this in the same form that we've looked at some of the other factoring problems can we think of two numbers whose product is 30x squared and whose sum whose sum is 11 X notice 11 X is the coefficient on Y we have Y squared some coefficient on Y and then in terms of Y some you know this isn't in any way dependent on Y so if you knew if you one way to think about this if you knew what X was then this would be a quadratic in terms of Y and that's how we're really thinking about it here so can we find two numbers whose product is 30x squared and two numbers whose sum is the coefficient on this Y term right here whose sum is 11x so let's just think about all of the different possibilities if we were just thinking about two numbers whose product was 30 and whose sum was 11 we would be thinking of 5 and 6 5 times 6 is 30 5 plus 6 is 11 and some trial and error you could have tried 3 & 10 well that would have been 13 would be their sum you could try 2 and 15 that wouldn't have worked you could you could but 5 & 6 does work here so we've already seen that multiple times so 5 & 6 5 & 6 would work for 30 but we have we have 30 x squared so what if we have 5x and 6x well 5x times 6x is 30x squared and 5x plus 6x is 11 X so this actually works so then our our factoring or our factorization of this expression is just going to be y plus 5 X y plus 5 X times y plus 6 X y plus 6 X and I'll leave it up to you to verify that this does indeed when you multiply it out equal this up here