I have a quick question here.
For an exercise, I was asked to factor:
$$11x^2 + 14x - 2685 = 0$$
How do I figure this out quickly without staring at it forever? Is there a quicker mathematical way than guessing number combinations, or do I have to guess until I find the right combination of numbers?
The answer is:
$$(11x + 179)(x - 15) = 0 $$
On
Here's one suggestion: Look at $11$ and $-2685$. Factorize $2685$ into primes to get $3.5.179$. Then check the following.
$11.(-3)+(895), 11.(-15)+179, 11.(3)+(-895), 11.(15)+(-179)$ and check which yields $14$. The one that yields $14$ is $11.(-15)+179$. So you get $(11x+179)(x-15)$.
I don't know if this method is standard but it seems to work for me.
You want to know the divisibility rules for small numbers. We know $2685$ is divisible by $5$ because of the last digit and by $3$ because of the sum of the digits. Once you find those factors, divide them out, getting $179$. The rules show it is not divisible by $2,3,5,11$ (or $7$ if you know that one, but it is less common. I like the double the ones digit and subtract, which just gives a yes/no answer). Since you only need to check primes up to the square root of the number, you would just have to trial divide by $13$ and maybe $7$ to find that $179$ is prime. In exercises there will always be small factors. In RSA encryption, not so much.