How many possibilities are there for b?

Question

An ancient manuscript contained this statement: "x^2 +bx + 30 has two integer roots." Unfortunately, it is impossible to read the positive integer b. How many possibilities are there for b?

Answer 1

30 * 1 ===> 31
15 * 2 =====> 17
10 * 3 ======> 13
6*5 =======> 11

Could be 11, 13, 17 , or 31

Four posiible values of b

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for the factors, we know they must be positve
I took all the factors of 30 and then we know they add to b

(x+6)(x+5) === 6*5 = 30 and we know 6+5 = 11 which will be b

Answer 2

30 = 2^1 * 3^1 * 5^1
(1+1)(1+1)(1+1) = 8 factors. Thus, there are 8 possible values of b.

Here they are if you want to see them:

1*30 -> 1+30 = 31
2*15 -> 2+15 = 17
3*10 -> 3+10 = 13
5*6 -> 5+6 = 11
(-1)(-30) -> -1 - 30 = -31
(-2)(-15) -> -2 - 15 = -17
(-3)(-10) -> -3 - 10 = -13
(-5)(-6) -> -5 - 6 = -11

In general, for: x^2 +bx + C, it will be the number of factors "C" has if C is not a square, and it will have one more than the number of factors if it is square. We can condense this into: 2⌈n/2⌉ where n is the number of factors of C, and ⌈x⌉ is the ceiling function.

(x+a)(x+c) = x^2+x(a+c)+ac

If C has n factors, then each factor of C pairs up with another, producing "n/2" pairs. Each of those pairs has a corresponding negative pair, as a*c = (-a)(-c). So, we can produce two solutions with every pair. (n/2)2 = n.

If C is a square, then one of the pairs will be itself, so there are (n+1)/2 pairs. Each pair also has a corresponding negative pair, for the same reason above. Thus, there are (n+1)/2*2 = n+1 solutions.

Answer 3

I can find four : 31, 17, 13 and 11. You need to find all factors of 30, and pair them up , then add the pairs to get the middle term.

I find 1 * 30 which adds to 31, 2 *15 which adds to 17, 3 * 10 which adds to 13, and 5 * 6 which adds to 11.

If you also did not know the sign for b (and it could be negative), you have four more possiblities -- the negatives of all these -- since they would still multiply to +30

Answer 4

Use the quadratic formula;

x = [-b +/- sqrt(b^2 - 4ac)]/(2a)

with a=1 and c=30 here. Now if it's going to have integer roots, this puts many restrictions on b. For instance, whatever is inside the sqrt had better be a perfect square. Trying playing around with this.

Answer 5

Yes it's possible that you are pregnant and that you also have a sexually transmitted disease. The pull out method will not stop you from getting pregnant. Many boys cannot control themselves. If you are concerned about pregnancy, pulling out (a.k.a., withdrawal) is not the most reliable or forgiving form of contraception. Re-inserting your penis after you've ejaculated (outside the vagina) is not safe either. Within the first few hours after ejaculation, there is usually some sperm left in the urethra. The amount left behind is nowhere near the amount of sperm in the typical ejaculate, but it still presents a risk. Pulling out, also called "withdrawal" and "coitus interruptus," is risky because the guy has to be totally aware of when he's going to ejaculate (***). Then, he has to have enough self-control to pull out before he cums, every single time. Just one mistake can result in an unplanned pregnancy. A lot of guys are unable to have this much control, especially when they're so close to the pleasure of orgasm and ejaculation. Plus, a woman can get pregnant even if a guy ejaculates just a little bit of semen near the opening of her vagina, because sperm can swim inside. Safer sex is not just about preventing unplanned pregnancy. It's also about keeping you and your partner healthy - and free from sexually transmitted diseases (STDs), which can be spread both through pre-*** and vaginal fluids. Pulling out is better than using no birth control at all. But it's unreliable and gives no protection against STDs. Are you really ready to take those chances? If your period is late next month you can take a home pregnancy test to find out if you are pregnant.

Answer 6

8. But now *you* have to figure out why ☺

HTH

Doug

Answer 7

few and make your own homework...dont be lazy...think logical for 5 minutes

Answer 8

4...11,13,17,31