cooo3 wrote:maybe i missed this somewhere but is there a certain rule how to factor out an equation?
i tried to solve x^2+12x-540=0 but i never would have guess that this equation is the same as(x-18)(x+30).
is there a certain rule or is it just practice?
any help is much appreciated.
There are definitely rules; I'll set the key ones out below, but you'll want to refresh your memory on quadratic equations in general.
"Reverse FOIL", as it's often called, is the way that we pack that long equation into two brackets.
The first place we look is the coefficient of the x^2 term; if it's 1 (i.e. there's no number there), our life is easy, we can put an "x" at the front of each bracket. If there's another number there, then factoring becomes more interesting. Good news - on the GMAT, that rarely happens.
So, for your equation, we now have:
(x )(x ) = 0
The next place we look is the sign before the number at the end.
If that sign is positive, then we know that the two signs in the brackets will be the same (since to get a positive product you need either two positives or two negatives). We then look at the sign before the "x" term to choose our sign.
(For example, if our quadratic had been:
x^2 + 5x + 6 = 0
the second + sign tells us that the signs are the same; the first + sign tells us that they're both positive. In this case we'd start with:
(x + )(x + ) = 0).
If that sign is negative, then we know that one bracket will have a + sign and the other a - sign (since you need a negative and a positive to generate a negative product). In your example we have "-540", so we need a positive and a negative:
(x + )(x - ) = 0
Now we look at the actual numbers. We want two numbers that multiply to 540 and subtract (because of the "-"540) to 12, the number in the middle term. Only 30 and 18 fit that bill.
Since each sign is different, we need to be careful where to put each number. We have a positive balance of "x"s (+12x is the middle term), so we need to generate a positive balnce. Accordingly, the bigger number goes with "+" and the smaller number with "-":
(x + 30)(x - 18) = 0
Finally, to solve for x we use a basic principle of "0": the only way to get a product of 0 is to multiply by 0. In this case, that means that:
x + 30 = 0
x = -30
OR
x - 18 = 0
x = 18
