One thing that you should know is that any quadrilateral's angles sum to 360. Thus, we can make the equation
(q) + (5q + 15) + (3q - 55) + (4q - 20) = 360
This is just simple algebra that you should be able to do. You can simplify this, and then once you find out what q is, figure out what the largest angle is.
BTW this question was also answered here: https://web2.0calc.com/questions/geometry_51921
We can simplify first:
x^2 + (b - 6)x + 14 = 0
Now we can use a bit of brute force here:
(x + p)(x + q)
pq = 14
p + q = b - 6
(p, q) = (14, 1) (7, 2) (-14, -1) (-7, -2)
So we can plug the values of p and q in to get all the possible values of b:
14 + 1 = b - 6 --> b = 21
7 + 2 = b - 6 --> b = 15
-14 + -1 = b - 6 --> b = -9
-7 + -2 = b - 6 --> b = -3
Thus, all the possible values of b are: 21, 15, -9, -3.
We can use complementary counting, which is, subtracting the possibilities that don't satisfy the condition from the total amount of possibilities.
For the first die, the probability that two dice don't show the same number is 1.
For the second die, the probability is 5/6, because one possibility was taken from the first die.
For the third die, the probability is 4/6 or 2/3 because two possibilities have been taken from the two dice already rolled.
Therefore, we see that the fourth and fifth have 3/6 (1/2) and 2/6 (1/3) as their probability.
We multiply these all together and we get
1 * 5/6 * 2/3 * 1/2 * 1/3 = 5/54.
(This question was also answered here: https://web2.0calc.com/questions/help-one-more-problem)
Let's organize our information.
_ _ 6
There are 9 * 10 ways to fill in the other two spots. (Because the first one can't be 0)
We check the first ones and we find that
Each time the number grows by 40.
So this is (999 - 136) / 40 = 863 / 40 = 21.575, but then you have to count 136, thus the probability that the number is divisible by 8 is 22/90 = 11/45.
According to Wikipedia, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
The first one is just a mess.
The second one fits all conditions.
The third one has a variable exponent.
The fourth one has a negative exponent.
Thus the only one that is a polynomial is the second one, or x^4 - 2.
We can simplify the inequality first:
Dividing by 4,
x + 2y < 5
Remember the problem says positive integers. Therefore, we can count the limited number of possibilties:
(1, 2) doesn't work because 5 is not smaller than 5.
That's it, because if any of the numbers were larger, x + 2y would be bigger than 5.
Thus the number of ordered pairs of positive integers that satisfy the inequality is 2.
If you know combinatorics, you will see that this is simply 11 choose 3.
For those of you who don't, the formula for choosing a number out of another number is n! / ((n – r)! r!), where n = number of items and c = how many you choose.
11 choose 3 = 11! / (11-3)! * 3! = 11! / 8! 3! = 11*10*9 / 3*2*1 = 11*5*3 = 165 ways.
(Alphabetical order does not matter here because you could always reorder them to be alphabetical and combinatorics do not count each combination more than one time)
First, let's simplify it:
|5a + 2| < 9
For an absolute value inequality, you split it into two - one is positive and the other is negative. (Remember to flip the sign in the negative one!)
5a + 2 < 9
5a + 2 > -9
For the first one:
5a < 7
a < 7/5
For the second one:
5a > -11
a > -11/5
Thus we have our range: -11/5 < a < 7/5
Therefore, in interval notation, our answer is (-11/5, 7/5).