Factoring quadratics with perfect squares
Factor 25x squared minus 30x plus 9. And we have a leading coefficient that's not a 1, and it doesn't look like there are any common factors. Both 25 and 30 are divisible by 5, but 9 isn't divisible by 5. We could factor this by grouping. But if we look a little bit more carefully here, see something interesting. 25 is a perfect square, and 25x squared is a perfect square. It's the square of 5x. And then nine is also a perfect square. It's the square of 3, or actually, it could be the square of negative 3. This could also be the square of negative 5x. Maybe, just maybe this could be a perfect square. Let's just think about what happens when we take the perfect square of a binomial, especially when the coefficient on the x term is not a 1. If we have ax plus b squared, what will this look like when we expand this into a trinomial? Well, this is the same thing as ax plus b times ax plus b, which is the same thing as ax times ax. Ax times ax is a squared x squared plus ax times b, which is abx plus b times ax, which is another. You You could call it bax or abx, plus b times b, so plus b squared. This is equal to a squared x squared plus-- these two are the same term-- 2abx plus c squared. This is what happens when you square a binomial. Now, this pattern seems to work out pretty good. Let me rewrite our problem right below it. We have 25x squared minus 30x plus 9. If this is a perfect square, then that means that the a squared part right over here is 25. And then that means that the b squared part-- let me do this in a different color-- is 9. That tells us that a could be plus or minus 5 and that b could be plus or minus 3. Now let's see if this gels with this middle term. For this middle term to work out-- I'm trying to look for good colors-- 2ab, this part right over here, needs to be equal to negative 30. Or another way-- let me write it over here-- 2ab needs to be equal to negative 30. Or if we divide both sides by 2, ab needs to be equal to negative 15. That tells us that the product is negative. One has to be positive, and one has to be negative. Now, lucky for us the product of 5 and 3 is 15. If we make one of them positive and one of them negative, we'll get up to negative 15. It looks like things are going to work out. We could select a is equal to positive 5, and b is equal to negative 3. Those would work out to ab being equal to negative 15. Or we could make a is equal to negative 5, and b is equal to positive 3. Either of these will work. If we factor this out, this could be either a is negative-- let's do this first one. It could either be a is 5, b is negative 3. This could either be 5x minus 3 squared. a is 5, b is negative 3. It could be that. Or you could have-- we could switch the signs on the two terms. Or a could be negative 5, and b could be positive 3. Or it could be negative 5x plus 3 squared. Either of these are possible ways to factor this term out here. And you say wait, how does this work out? How can both of these multiply to the same thing? Well, this term, remember, this negative 5x plus 3, we could factor out a negative 1. So this right here is the same thing as negative 1 times 5x minus 3, the whole thing squared. And that's the same thing as negative 1 squared times 5x minus 3 squared. And negative 1 squared is clearly equal to 1. That's why this and this are the same thing. This comes out to the same thing as 5x minus 3 squared, which is the same thing as that over there. Either of these are possible answers.