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Factoring perfect squares: negative common factor

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We need to factor negative 4t squared minus 12t minus 9. And a good place to start is to say, well, are there any common factors for all of these terms? When you look at them, well these first two are divisible by 4, these last 2 are divisible by 3, but not all of them are divisible any one number. Will, but you could factor out a negative 1, but even if you factor out a negative 1-- so you say this is the same thing as negative 1, times positive 4t squared plus 12t plus 9-- you still end up with a non-one coefficient out here and on the second degree term, on the t squared term. So you might want to immediately start grouping this. And if you did factor it by grouping, it would work, you would get the right answer. But there is something about this equation that might pop out at you that might make it a little bit simpler to solve. And to understand that, let's take a little bit of a break here on the right hand side, and just think about what happens if you take a plus b times a plus b, if you just have a binomial squared. Well you have a times a, which is a squared. Then you have a times that b, which is plus ab. Then you have b times a, which is the same thing is ab. And then you have b times b, or you have b squared. And so if you add these middle two terms, right here, you're left with a squared plus 2ab plus b squared. This is the square of a binomial. Now, does this right here, does 4t squared plus 12t plus 9 fit this pattern? Well the 4t squared is a squared. So this right here is a squared. If that is a squared right there, then what does a have to be? If this is a squared, then a would be equal to the square root of this. It would be 2t. And if this is b squared, let me do that in a different color. If this right here is b squared, if the 9 is b squared, right there, then that means that b is equal to 3. It's equal to the positive square root of the 9. Now, this number, right here-- and actually it doesn't have to just be equal to 3, it might have been negative 3 as well. It could be plus or minus 3. But this number here, is it 2 times ab? Right? That's the middle term that we care about. Is it 2 times ab? Well if we multiply 2t times 3, we get 6t. And then if we multiply that times 2, you get 12t. This right here, 12t, is equal to 2 times 2t times 3. It is 2 times ab. And if this was a negative 3, we would look to see if this was a negative 12, but this does work for positive 3. So this it does fit the pattern of a perfect square. This is a square of a binomial. So if you wanted to factor this-- the stuff on the inside, you still have that negative 1 out there, the 4t squared plus 12t plus 9-- you could immediately say, well that's going to be a plus b times a plus b. Or 2t plus 3 times 2t plus 3, or you could just say, it's 2t plus 3 squared. It fits this pattern. And, of course, you can't forget about this negative 1 out here. You could have also solved it by grouping, but this might be a quicker thing to recognize. This is a number squared. That's another number squared. If you take each of those numbers that you're squaring, take their product and multiply it by 2, you have that right there. So this is a perfect square.