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# Factoring two-variable quadratics: rearranging

Sal rearranges 30x^2+11xy+y^2 as y^2+11xy+30x^2 and then factors it as (y+5x)(y+6x). Created by Sal Khan.

Video transcript

Let's see if we can use our
existing factoring skills to factor 30x squared
plus 11xy plus y squared. And 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 open up
what's going on here-- is to maybe rearrange
this a little bit. We could rewrite
this as y squared plus 11xy plus 30x squared. And my whole
motivation for doing that-- there are ways to
factor 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 got us 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 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 is 11x? Notice, 11x is the
coefficient on y. We have y squared,
some coefficient on y. And then in terms
of y, this isn't in any way dependent on y. So 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. It's some trial and error. You could have tried 3 and 10. Well, that would have been--
13 would be their sum. You could have tried 2 and 15. That wouldn't have worked. But 5 and 6 does work
here, so we've already seen that multiple times. So 5 and 6 would work for
30, but we have 30x squared. So what if we have 5x and 6x? Well, 5x times 6x is 30x
squared, and 5x plus 6x is 11x. So this actually works. So then our factoring
or our factorization of this expression
is just going to be y plus 5x times y plus 6x. And I'll leave it up to you to
verify that this does indeed, when you multiply it
out, equal this up here.