in this video we're going to focus on factoring perfect square trinomials so let's try an example let's see if we have the expression x squared plus 6x plus 9 how can we factor this particular expression what can we do well let's focus on the line what two numbers multiply to 9 but add to the middle coefficient 6 3 times 3 is equal to 9 but 3 plus 3 is equal to 6 so to factor this expression it's going to be X plus 3 times X plus 3 and you could check it out you can test it to make sure that this works to prove it oil the expression X plus 3 times X plus 3 x times X is x squared x times 3 is 3x and 3 times X is also 3x and finally 3 times stream is 9 so if you add the two middle term 3x plus 3x is 6x so you get the original expression x squared plus 6x plus 9 so we know this is the answer but we can write it as X plus 3 squared which makes it a perfect square trinomial so how do we know if a trinomial is a perfect square focus on me the first coefficient and the constant term the square root of 1 is 1 and the square root of 9 is string if you multiply these two numbers you're going to get 3 and if that result gives you a number that's if it's half of the middle term and then it's going to be a perfect square trinomial sometimes you might find this one to be useful if you have the form a squared plus two a B plus B squared then you can factor it by writing this a plus B to the second power so in our particular example this is x squared plus 6x which is really two times three X plus 9 or nine is the same as three squared and this is equal to X plus three squared let's try another example x squared plus 10x plus 25 let's factor this expression so is it a perfect square trinomial well let's find out the square root of one is one and the square root of 25 is 5 if we multiply one in five we're going to get five and five is half of 10 so that means it's a perfect square trinomial number two numbers multiply to 25 but add to the middle term 10 5 times 5 is 25 and 5 plus 5 is 10 so this is going to be X plus 5 times X plus 5 or simply X plus 5 squared so that's one way in which you can get the answer but let's use the formula a squared plus 2 a B plus B squared is equal to a plus B squared so notice that a squared is x squared and B squared is 25 which means B is 5 so I can write 25 as being 5 squared plus 2 times a which is X and B which is 5 so as you can see - a B is 2x times 5 which is 10 X - equivalent to each other so therefore we know the answer is going to be X which is a plus B or 5 squared you don't have to use the formula but you could if you want to what about this one 4x squared plus 12x plus 9 is it a perfect square trinomial well let's find the square root of 4 and the square root of 9 the square root of 4 is string I mean not threes I take that back it's 2 the square root of 9 is string and if we multiply 2 times 3 this is equal to 6 which is half of 12 so that tells us that this is indeed a perfect square trinomial so we know that a squared is going to be 4x squared which means a is 2x and B squared is 9 so be a string and we know the answer is going to be a plus B squared so that means it has to be 2x plus 3 squared so according to the formula this is the answer and it's factored form let's prove it so because the leading coefficient is not 1 the techniques that we'll need to use to factor this expression it's going to be a little different than the last example now the first thing you need to do is multiply 4 + 9 4 times 9 is 36 now what two numbers multiply to 36 but add to the middle coefficient 12 this is going to be 6 + 6 6 times 6 is 36 6 plus 6 is 12 now let's replace 12 X with 6 X + 6 X and then we need to factor by grouping in the first two terms take out the GCF which is 2 X 4 x squared the by 2 X is 2 X 6 X divided by 2 X is stream now the GCF for the last 2 terms is stream 6x divided by 3 is 2 X 9 divided by 3 is 3 so we can factor out 2x + 3 so if we take out 2x + 3 we'll left over with 2x and if we take it out here what we have left over is 3 so as you can see 2x plus 3 times 2x + 3 is simply 2x + 3 squared now let's try another example let's say if we have the expression 9a squared plus 30 a B plus 25b squared is this expression is it a perfect square trinomial well let's find out what is the square root of 9 and the square root of 25 the square root of 9 is 3 and the square root of 25 is 5 and 3 times 5 is 15 and that is that 15 is half of 30 which means that this is indeed a perfect square trinomial so a squared is going to be equal to the first part of the bottom of this expression and that is uh 980 squared so a is therefore the square root of 90 squared which is 3/8 B squared is equal to the last part of this trinomial that's 25 lowercase B squared so capital B is going to be 5 lowercase B so we know the answer is going to be a plus B squared so therefore the final answer should be 3a plus 5b squared if it's a perfect square trinomial this is what you can do to quickly get the answer so it should be 3a plus 5b squared but now let's confirm it so first we need to multiply the leading coefficient and the last coefficient 9 times 25 is 225 what two numbers multiply to 225 that adds to the middle term 30 15 times 15 is 225 and 15 plus 15 adds up to 30 so we're going to replace the middle term 30 a B with 15 a b + 15 18 now let's factor by grouping in the first two terms take out the GCF the greatest common factor is 3 a 90 squared divided by 3 a is 3 a + 15 18 divided by 3 a is 5 B now let's take out the GCF in the last two terms which is 5 B so 15 a B divided by 5 B is 3a 25b squared divided by 5 B is 5 B so now let's factor out 3a plus 5b so if we take out 3a plus 5b in the first expression or the first term we have 3a left over and if we take out doing a plus 5b in the last one we have five B left over so 3a plus 5b times 3a plus 5b is 3a plus 5b squared