1. x to the second power + 6x +9 2. 4x to the second power +20x + 25 3. 36x to the second power - 24x +16

Question

determine whether each trinomial is a perfect square. if so, factor it, if not, explain why

Answer 1

?????????? did yu tri tu sae "x^2 +6x+9.2=0" ???????????

quadratik equashun giv roots=-3+-0.447213595i (komplex variabels)

b^2-4ac=-0.8, so sqrt(-0.8) giv 0.894427191i

Answer 2

1. x^2+6x+9. 1 is the coefficient of the x^2 term so we know we will have x in each factor bracket when we factorise. If the x coefficient halved and then squared is the constant term, then it's a perfect square. So we have (x+3)^2.
2. 4x^2+20x+25. This time the coefficient of x^2 is 4, which means we could have 4x in one bracket and x in the other; or we could have 2x in each. The factors of 25 are 1 and 25 or 5*5 and we know that if it's a perfect square, the middle term must be 2*(2*5)=20, which it is! So the perfect square is (2x+5)^2.
3. 36x^2-24x+16 (=4(9x^2-6x+4) could be a perfect square because 36 and 16 are the squares of 6 and 4 (or 3  and 2). If the middle term had been 2*(6*4)=48 (or 12, if we take out the common factor 4) which it isn't, then we would have had a perfect square. Uh-oh! It's not a perfect square but does it factorise? Here's a way to do it: the middle term is 2*(6*X)=24, where big X is the other square besides 6. So X=24/12=2. (If we work with 9x^2-6x+4, we have 2*(3*X)=6, so X=1.) So the perfect square is (6x-2)^2=36x^2-24x+4 (or (3x-1)^2=9x^2-6x+1). We have to adjust the original 16 (or 4) to make up for the 4 we just put in so we have 12 (or 3) and the expression becomes (6x-2)^2+12 (or (3x-1)^2+3) which doesn't factorise. Why not? If we equate the expression to zero, then we could solve for x and discover the factors. If we equate to zero we get (6x-2)^2=-12 (or (3x-1)^2=-3). In other words, we have a number whose square is negative, which isn't possible in the real world. So there's no (real) solution, so we can't factorise the original expression in terms of x, and we can only simplify it by dividing by the common numerical factor 4: 4(9x^2-6x+4).