so i only need to concentrate on the figures to the far right, in this case, the 7 and 12? (it's things like this which are going straight over my head ) and is it always the number on the end that needs to be obntained by multiplication?
If you look at a factorised quadratic equation:
(x + 4)(x - 2)
You can find out how they work. Since we don't know what x is, we can't multiply the two sides together to get a result, we have to multiply each component each other one.
So for example:
in (x + 4)(x - 2) you have:
x * x = x²
4 * x = 4x
x * -2 = -2x
4 * -2 = -8
When you add this all up, you get:
x² + 4x - 2x - 8
OR
x² + 2x - 8
So you can see that the 2x in the equation is made by adding the two numbers in the factorised equation together (in this case, 4 and -2 to give us 2), and the last number is made by multiplying the two numbers together (4 and -2 to give us -8).
This means in order to factorize the quadratic equation (let's say: x² - 6x - 16), we need to get two numbers, that multiply to make -16, and add together to make -6. The factors of -16 are 1 * -16, 2 * -8, 4 * -4, 8 * -2, and 16 * -1. If we look at these combinations, which can we add together to get -6? -16 and 1 gives us -15. 4 and -4 give us 0, 8 and -2 give us 6, 16 and -1 give us 15, but 2 and -8 give us -6, which means we just need to substitute that into the (x )(x ).
So in this case: (x + 2)(x - 8). We can see this is right by splitting it up and multiplying each component:
x * x = x²
x * -8 = -8x
2 * x = 2x
2 * -8 = -16
x² - 8x + 2x - 16 = x² - 6x - 16
It gets a little trickier when we get things like "2x²" or "6x²". To make these, you need to multiply x's in the brackets. 2x² is easy. The only combination that multiply to make that is x and 2x. The problem comes, however, when putting these into the factorised equation, because the 2x is multiplied by the number too. So for example:
(2x + 5)(x + 3)
gives us:
2x * x = 2x²
2x * 3 = 6x
5 * x = 5x
5 * 3 = 15
Which, added up, is : 2x² + 6x + 5x + 15
Simplified: 2x² + 11x + 15
To factorise that quadratic, we do the same as the one above, but consider that one of the sides has to have a 2x on it, and compensate for that.
If we look at the factors of 15:
1 and 15
3 and 5
Neither of those can add up to make 11. But we have to consider the 2x, which means one of the numbers will be multiplied by that. So we could have:
2x + 15x
1x + 30x
6x + 5x
3x + 10x
Now we can see that one of those makes 11x (6x + 5x). This is made by multiplying the 5 by x, and 3 by the 2x. Remember that the multiplied ones are in the opposite brackets to each other, so we get:
(2x + 5)(x + 3)
It gets worse with different x²'s though. For example: "4x²". This can be made with both 2x * 2x and x * 4x. This is the same principle as the the other one, but we need to consider all the combinations for 4x².
So, say we have the quadratic: 4x² + 19x + 12
We start off by finding the factors of 12:
1 and 12
2 and 6
3 and 4
Our combinations for 4x² are x * 4x, and 2x * 2x. So we find every combination for every factor of 12:
1 and 12 gives us:
1x and 48x (49x)
4x and 12x (16x)
2x and 24x (26x)
2 and 6 give us:
2x and 24x (26x)
8x and 6x (14x)
4x and 12x (16x)
3 and 4 give us:
3x and 16x (19x)
12x and 4x (16x)
6x and 8x (14x)
If we look at the quadratic equation again, we need 19x in the middle. As you can see, the only combination here that gives 19x is 3x and 16x (which is made from 3 * x, and 4 * 4x). Using this, we can figure out the factorised equation. Again, remember that the multiplied ones are in the opposite brackets. So:
(4x + 3)(x + 4)
is our answer.
It looks very long, and it is, but once you get better, you can start to get a good idea in your head of what the answer will be, and you will get faster at doing them.
Negative numbers can be annoying but remember:
if the last number is subtracted (x² + x
- 2) one of the sides of the factorised equation must be a minus (x + a)(x - b)
if the last number is added (x² + 5x
+ 3), both sides of the factorised equation must be additions or subtractions (x + a)(x + b) or (x - a)(x - b). We can tell which one to use, by the middle number. If it is added (x²
+ 4x + 2) it's (x + a)(x + b) OR (-x - a)(-x - b). If it is subtracted (x²
- 4x + 4), then the factorised equation must follow (x - a)(x - b) OR (-x + a)(-x + b)
I probably haven't described that very well and have only confused you more, but I would advise getting a GCSE maths textbook or something. It'll explain it much better than I can.
If you have any questions, post them here, and I'll show you how I work them out.
Also, if you can remember jotun's formula, that's the best and easiest way to work them out. I can't, so I use the old fashioned method. A calculator with that function built in would be very useful, but I would still advise knowing how to do it anyway. If they want you to work them out, I'd wager that they'd also expect you to explain how you worked them out.
