Even though I think that I might be too late.. (which I hope I'm not), I'm going to try to help, because it might be better for the long term anyway.
I skipped certain sections because all of the problems in them were correct.
Factor as a Binomial Square
You can only factor as a binomial square when the roots are a difference of each other. Otherwise, such as in the case of the second problem, you have to use the quadratic formula, because the roots are imaginary.
You did the first problem correctly, except that 7 is not the square root of 40. (I'm assuming that you meant 49)
Factor as a Trinomial Square
When you factor as a Trinomial square, you're roots have to be the same. The second one is correct, the first one is not. (All you have to do for the first one is change one of your signs.)
Solve
9x^2-25=0
I'm going to go through this one. You're objective at first is to get x^2 by itself, so you can put it under the radical and get rid of the square.
1. Subtract 25
2. Divide 25 by 9
3. x^2=25/9 <-- put both sides under the radical. 25 and 9 are both
perfect squares, so you get 5/3.
In problems such as 2-4, you need to set them equal to zero and factor. Then you set your roots equal to zero and solve.
Simple ex. x^2+12=-7x
x^2+7x+12=0 (I automatically put them in decreasing power order, it makes it easier)
(x+3)(x+4)=0
x+3=0 x+4=0
x=-3 x=-4
The last problem is almost correct, but remember, when you find the radical of anything, its both is positive and negative value.
Factor
For the first problem, you have the right idea. But, the objects in the parentheis (c-7),(7-c), need to be the same. To do this, you need to factor out a -1 in one of them:
5a(c-7)-3(7-c)
5a(c-7)-3(-1)(c-7)
5a(c-7)+3(c-7)
(5a+3)(c-7)
The second and third problems are right so far, but since (x^3+8 ) and (x^2-9) are still factorable, you just need to factor them too.
For the last problem, you are almost right. When you distribute, you only need to factor out one value for your binomial. (i.e. if you have 2x+4, and you chose to factor out a 2, your answer will be 2(x+2) rather than (2+2)(x+2) ).
I really hope that this helps, feel free to PM me if anything that I said is confusing and I'll try to explain it more.
I'm also super sorry if I'm too late.
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