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The perimeter is 4*sqrt(3).  And the area is 2*sqrt(3).

The maximum value of PQ + F_1 Q is 8.

No explanation, and it's wrong.

For a quadratic to have two real roots, the discriminant must be greater than 0. So we require

\(\begin{align*}15^2-4 \cdot 8 \cdot c &> 0 \\ \Rightarrow \quad 225-32c &> 0 \\ \Rightarrow \quad c&< \frac{225}{32}.\end{align*}\)

The largest integer smaller than \(\frac{225}{32}\) is 7. Thus, the positive integer values of c are 1, 2, 3, 4, 5, 6, and 7 and their product is \(\boxed{5040}\).

(a,b) = (1,sqrt(3)).

The possible values of c are 1, 2, and 3, so the answer is 1*2*3 = 6.

[3^2004 * 5^2004] mod 10^10 ==5087890625 - these are the last 10 digits.

234696

234696 / 2 ==117,348

234696 / 3 ==78,232

234696 / 7 ==33,528 

MATH has a "period of 4"

Therefore: 1000 mod 4 ==0, which means the last letter of MATH, or H

Distinct odd-digit numbers. It means only numbers that comprise of odd digits only.

[(1,), (3,), (5,)] [(1, 3), (1, 5), (3, 1), (3, 5), (5, 1), (5, 3)] [(1, 3, 5), (1, 5, 3), (3, 1, 5), (3, 5, 1), (5, 1, 3), (5, 3, 1)] >>>Total distinct permutations with odd digits only == 15

See this pattern:

M - 1, 5, 9, 13, 17...

A - 2, 6, 10, 14, 18...

T: 3, 7, 11, 15, 19...

H: 4, 8, 12, 16, 20...

Do you think you can use this information to extrapolate the 1000th position? 

Edit: Think about 997, 998, 999, 1000. Where will these fall?

this is a duplicate:
https://web2.0calc.com/questions/help_33122

OP, try adding the two equations and see if you can find a value for b in terms of a. Then substitute that value back into the system.

79 , 88 , 97 , 169 , 178 , 187 , 196 , 259 , 268 , 277 , 286 , 295 , 349 , 358 , 367 , 376 , 385 , 394 , 439 , 448 , 457 , 466 , 475 , 484 , 493 , 529 , 538 , 547 , 556 , 565 , 574 , Total Num = 31 words in her birthday message.

If you are OP, the first answer by that Guest was most likely a troll. I will reitirate what I said in my answer - this problem is very basic. You didn't even take the time to check if the Guest's answer was right, which means you are literally just here for a shortcut to your homework. I recommend actually attempting the problem before coming here.

If you are still having issues, I have posted some hints to the process of this equation in my answer (#2), hope it helps. 

The values of a are 8 and -8.

It says its not correct

 I am going to be blundt- this is a pretty simple question. I recommend looking at YouTube or Khan Academy videos on simulataneous linear equations if you are having trouble with this.

Try adding the two equations, solving for x, and plugging your x value back into one of the equations.

Please try and digest my process, or you won't learn anything. Happy trails!

(x,y) = (8,7)

Hint 1: Don't think about the words as words. Just remember that there are 8 2-syllable words and 7 3-syllable words, and they are all unique. Make sure to correct for overcounting with this information!

Hint 2: The first and last lines can only have one 2-syllable and one 3-syllable. The second line can have two 2-syllables, and one 3-syllable. Use this to your advantage.

Note that there are 5 choices for the final digit. 

Of these, 2 are odd, so the probability is \(\color{brown}\boxed{2 \over 5}\)