Let's talk Quadratic Equations

I should have said "best way to torture students", but either would work.

So our school has a program called A+. Very simple program to understand. You take a computerized test over various math formulas to determine what your strong and weak points are. The test will then determine which area of math you have trouble on (as in, specifics. Fractions, dividing fractions, etc etc). Each induvidual math lesson comes with three steps: Study guide, practice test, mastery test. Both tests are 10 questions long, and you have to get at least 80% to pass. Once you pass the mastery, you can move onto another lesson. Simple.

I have been on the Quadratic Equations lesson for three months. To say I'm sick and tired of it would be the understatement of the century. My math teacher is not helping at all, as he is an insistant git, but moving on. I ask him for help on the aspects I have trouble with and he refuses to help me. He points to my notes and tells me to figure it out myself. I am one very desperate person right now. If I don't pass the other lessons I won't get the math credit, and therefore, might not graduate.

I can SOLVE the problems. I don't have trouble with that. What I do have trouble with, however, is figuring out wtf the answers would be..positive or negative? He's explained it to me and I simply don't understand. And I feel really dumb because I already learned this in 7th grade. As you can tell, math is not my forte. So here's a random Quadratic Equation:

2x + 4 - 30 = 0

First problem: When writing the factors, what would come first? The positive or the negative sign? (x + a)(x - a)? or (x - a)(x + a)? And once I DO find the factors, how do I know which one would be positive and which one negative? -x and x? Also, which factor would go in the negative space? Augh. Any help on this would be lovely. Thanks.

Yes, the signs do get confusing - there are two ways of solving a problem like this. In your problem, (the 2 in the parentheses means squared, I can't do superscript on vBulletin) 2x(2) + 4x - 30 = 0, you could save yourself some trouble and divide everything by two, but I don't know if this is standard. I'd just divide out what I could.

At any rate, in 2x(2) + 4x - 30 = 0, in ax(2) + bx + c = 0 form, you understand how a and c have to be put together in a way to make the middle term. You know this, so I'll skip the discovering that step.

Now we know that a, which is, in this case 2, is 1 and 2, as that is the only multiplication duo (in whole numbers) that will work. As well, using deduction, we know that c, 30, is 3 and 10. So we have four numbers:

a.....c
1.....3
2.....10

We know that a is either both positive or both negative because it has to equal a positive 2, and we know that in c one has to be positive and the other has to be negative because it has to equal a -30, and so it works out like this:

..a.......c
+1.....-3
+2.....+10

We know we have to make 2 and -3 go together to get -6, and we know that 1 and +10 have to go together to get 10, as -6 + 10 = 4 (b). As we both know, this is where it gets confusing, as the ones we are multiplying go like so:

..a.......c
+1.....-3
......X
+2.....+10

But, here is the key, do a diagram like this, illustrating which two you want to multiply. But, the ones that, when multiplied together, makes sense - are backwards. When you do this, draw lines from illustrating which ones make sense, like the above, with the 2 to the -3 and the 1 to the 10. Then, you do the opposite. You set it up so that it is 2 is with the 10 and 1 is with the -3, getting (2x +10)(x - 3)

Was that too complicated? I could illustrate the other way but its more complicated (but much less confusing - I never use the above method anymore, I use the complicated one) and I can't make vBulletin do square root signs. If this didn't help you, I'm sorry. Just keep in mind that a and c's two factors have to equal the middle number when multiplied together and added.

The order of the positives/negatives doesn't matter.

As for which one is positive/negative, that depends how you're factoring it.

If your equation was, say x^2+5-45 = 0 then you'd do it like this. (The 2x is a needless complication at this point, also, I'm assuming you mean x^2, not just x and that you meant -4x, not just -4.)

You need to find two numbers that add to 4, but multiply to -30.

In this case, those numbers are 9 and -5.

So, when you make your factors, you simply put the numbers into the brackets.

So, that equation factored is (x+9)(x-5).

If you aren't sure of your answer then it's easy enough to check, just multiply those brackets out.

x*x = x^2
x*-5 = -5x
-9*x = 9x
9*-5 = -45

Put those together and you get: x^2 -5x + 9x -45 = 0
Simplify that a bit, and you get: x^2 +4x - 45 = 0.

Since that equals the equation you started with it's factored correctly.

I just read all that.. oww. I'm so glad I never did higher level math o.O;.

If you ever need to remember the quadratic formula, just keep in mind that it can be sung to the tune of "Pop Goes the Rainbow":

x equals negative b, plus or minus the square root of
b-squared minus 4ac, all over 2a

...sheez, that should be in the next Stephen King novel just for sheer creepiness factor.