Factoring perfect squares

factor 25 x squared minus 30x plus 9 and so 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 is it divisible by 5 so 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 so 25 x squared is a perfect square it's the square of 5 X and then 9 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 5 X so maybe just maybe this could be a perfect square so 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 so if we have a X 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 a B X plus plus B times ax which is another you could call it b ax or a b x plus b times v so plus b squared so this is equal to a squared x squared plus these two are the same term plus 2 a B X plus B squared so this is what happens when you square it a binomial now this pattern seems to work out pretty good let me rewrite our problem right below it we have 25 x squared minus 30x plus 9 so 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 and that means that the let me do this in a different color the B squared part is 9 that tells us that a that a could be a could be plus or minus five and that B could be and that B could be plus or minus three now let's see if this gels with this middle term if these four this middle term to work out for this middle term to work out I'm trying to look for good colors to a B this part right over here to a B needs to be equal to negative 30 or another way let me write it over here to a B needs to be equal to negative 30 or if we divide both sides by 2 a B needs to be equal to negative a B needs to be equal to negative 15 so that tells us since their product is negative 1 has to be positive and one has to be negative now lucky for us the product of 5 & 3 is 15 so if we make one of them positive and one of them negative will get up to negative 15 so it looks like things are going to work out so we could select we could select a is equal to positive 5 and B is equal to negative 3 those will work out those would work out to a B being equal to negative 15 or we could make a is equal to negative 5 and B is equal to positive 3 so either of these will work so if we factor this out this could be either a is negative this first one it could either be a is 5 B is negative 3 so this could eat either be 5 X minus 3 squared a is 5 B is negative 3 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 5 X plus 3 plus 3 squared so either of these are possible ways to factor this term out here and you're to 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 square and that's the same thing as negative 1 squared times 5x minus 3 squared and negative 1 squared is clearly equal to 1 so that's why this and this are the same thing this this comes out to the same thing as 5x minus 3 squared which is the same thing as that over there so either of these either of these are possible answers