idyllic

first, we should calculate the values of f(-2) and f(2) to be able to simplify the equation.

-2 is less than 0, so we're using the first function.

f(-2) = (-2)^2 - 2 = 4 - 2 = 2.

2 is greater than 0, so we're using the second function.

f(2) = 2(2) - 28 = 4 - 28 = -24

now let's plug it back into the equation and solve for a.

2 - 24 + f(a) = 0.

-22 + f(a) = 0

f(a) = 22.

since we now know what f(a) can be, we can work backwards using the two given functions.

let's start with the first function.

n^2 - 2 = 22

n^2 = 24

n = -sqrt(24) , as it's the only value less than 0 that satisfies the equation. (THOUGH: i'm not completely sure about this...)

second function,

2n - 28 = 22

2n = 50 

n = 25

so we get our difference to be 25 - (-sqrt(24)) which is 25 + sqrt24, or 25 + 2sqrt6.

i'm not completely sure about this answer though, hopefully it still helped you get some general idea of the problem - please let me know if you have any follow-up questions! 

you can notice that the arithmetic sequence increases by 2/3 each term.

in order to find the 8th term, we do 

2/3 + 7 * 2/3

= 2/3 + 14/3 

= 16/3

for future reference, the general formula for finding the nth term of an arithmetic sequence is: \(a_n = a_1 + (n-1)d\)

hope this helped! please let me know if you have any further questions :)

when you're trying to find the greatest product, you want to have two numbers that are the closest possible distance to each other.

in this case, the pair would be 12 and 12, so the greatest product is simply 144.

to verify this answer, you can try out different pairs.

for example, let's try 13 and 11. 

13 * 11 = 143, which is less than 144. you can continue trying other pairs if you are still doubtful ;)

hope this helped! please let me know if you are confused about anything i did 

i'm not CPhill, but here is my take on this problem:

first, let's expand and simplify the expression - 

pq + 3q + 4p + 12 - (pq + q + 2p + 2) = 44

pq + 3q + 4p + 12 - pq - q - 2p - 2 = 44

2q + 2p + 10 = 44

2q + 2p = 34

and when we divide both sides of the equation by 2, we get p + q = 17.

hope this helped! please let me know if you are confused about anything i did 

when we expand (Ax+1)(Bx+5), we get -

ABx^2 + 5Ax + Bx + 5

we know that the coefficient of x^2 in the initial expression is 6, so therefore we know AB must be 6.

hope this helped! please correct me if i am wrong :)

this way is probably not the most efficient way to do it, but i just listed out all the multiples of 8 that are between 1 and 100.

here is the list:

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96.

when you count, there are 12 multiples of 8.

so the answer is 12/100, or 3/25. 

another method to solve this is to simply do 100 / 8, which is just 12 remainder 4.

since the quotient is 12, that means there are 12 multiples of 8 in between 1 and 100.

hope this helped! please let me know if you are confused about anything i did 

check out the solutions hectictar and Melody have provided here:

first, let's add x to both sides to get:

2x - 5 + x < -12

3x - 5 < -12

now let's add 5 to both sides.

3x < -12 + 5

3x < -7

lastly, let's divide 3 from both sides.

x < -7/3

so, in interval notation, the answer would be:

(-inf, -7/3)

hope this helped! please let me know if you are confused about anything i did :)

to solve the perimeter of this trapezoid, we'll first need to calcuate the value of the height. we can do this by using the Pythagorean Theorem.

(19 - 13)^2 + h^2 = 10^2

6^2 + h^2 = 100

36 + h^2 = 100

h^2 = 64

h = 8

so now we can calculate the perimeter.

to do this, we simply just add 13 + 10 + 19 + 8 to get 50, and we're done!

hope this helped! sorry if my explanations weren't too clear. please let me know if you are confused about anything i did 

check out CPhill's answer here:

the formula for the area of a triangle is length * width, or perhaps base * height.

i'm not really sure how to explain this, but here is a site that may explain it better:

check out CPhill's answer here:

on the original post, Melody has given some great hints. i hope you can learn to understand them better in the future!

well, following Melody's hints, we can solve the problem (the steps in bold were not by me, they were by Melody):

1) find angle PST in terms of x                           180-(2x+8)      

2) Find angle PTS in terms of x                            x

3)  Solve for x                                                       we get this equation: 40 + x + 180 - 2x - 8 = 180  (this is because all angles in a triangle sum up to 180).

                                                                               212 - x = 180

                                                                               - x = -32

                                                                               x = 32

4)  Determine STR                                                angle STR is on the same line as angle PTS, which is simply just x. so angle STR is 180 - 32, which is 148 degrees.

hope this helped! please let me know if you have any follow-up questions