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\(\text{The letters A through G can be coded as 0 through 6}\\ \text{and thus a 3 letter set is equivalent to a 3 digit base 7 number}\\ \text{There are }7^3=343 \text{ such numbers and thus 343 sets of initials}\)

Nice, Jess!!!

That should do it, Android!!!!!

x^2 - x - 2

Let's find the zeroes (where the function crosses the x axis)

Factor as

(x - 2) ( x + 1)  = 0

Set each factor to 0  and solve for x

x - 2 = 0                             x + 1 =  0

 x = 2                                    x = -1

Notice that the graph on the bottom left appears to be correct  !!!!

\(\text{It should be pretty clear that the singularity occurs when }x=-7\\ \text{That the problem identifies it as a hole means that }x^2+3x-28 \\ \text{Should be divisible by }x+7\\\)

\(x^2+3x-28 = (x+7)(x-4) \text{ and thus}\\ \dfrac{x^2+3x-28}{x+7} = x-4,~x\neq -7\\ \text{thus the hole appears at}\\ (-7,~-7-4)=(-7,-11)\)

A rectangular cow pasture is enclosed on three sides by a fence and the fourth side is part of the side of a barn that is 400 feet long. The fence costs $5 per foot and $1200 altogether. To the nearest foot, find the length of the side parallel to the barn that will maximize the area of the pasture.

Dividing  1200 by 5 tells us how many feet of fencing we can afford.....so  ....1200 / 5 = 240 ft

Let the side of the fence paralleling the barn be x feet

Then....one of the other sides  perpendicular to the barn can be represented as   (240 - x) / 2

So....the area to maximize is

A =  x (240 - x) / 2       simplify

A = (240x - x^2)/ 2

A = (-1/2)x^2 + 120x

The x  that will maximize this is given by :      -(120) / (  2 * (-1/2) )  =  120 ft

And this is the length of the fence parallel to the barn that will max the area

3.

Similarly to question 2, you should simplify all the like terms first

x^2+12x+61=0

in ax^2+bx=c you get a=1 b=12 c=61

use the quadratic formula to find the roots of the equation

solve this and you should get the 2 roots

x= -6+5i , x= -6-5i

4.Find all real solutions to x^2+4=100x^2+20x+1.

Rearrange as

99x^2 + 20x - 3   =  0

Using the quad formula.....we have

( -20 ±√ [ 20^2 - 4 * 99 * -3 ]  ) / (2 * 99)

( -20 ±√ 1588) / 198

(-20 ± 2√397 )/198 =

(-10 ±√397 ) / 99   .....and these are the two solutions

Thank you to both of you :D

Don't know about that....LOL!!!!

no problem!

you are much better at explaining than me!

Thanks, Jess !!!!

2.Find the smallest real value of x such that x^2+6x+9=24.

Factor the left side

(x + 3)^2  = 24               take the negative root  [ since we want the smallest value ]

x + 3 = - √24       subtract 3 from both sides

x =  -√24 - 3  =    -2√6 - 3      and this is the smallest real value that makes the equation true

2. 

move all the numbers on one side so that the right side is left with 1

x^2+6x-15=0

use the quadratic formula to find smallest x can be

you should get something like

x = -3-2sqrt(6)

There are many things in between those two values, it can go on for infinity with numbers like 0.00000000000000000000001,0.00000000000000000000002, etc. but straight in the middle is 1/4

1. What is the largest value of k such that the equation 6x-x^2=k has at least one real solution?

Rearrange as    x^2 - 6x + k = 0

If this has at least one real solution, the discriminant must be ≥ 0

So

(-6)^2 -   4(1)(k) ≥ 0

36 - 4k ≥ 0

36 ≥ 4k 

9 ≥ k

So .....the max value  of k that produces at least one real solution is when k = 9

there is a very simple solution to this:

2,000/25= 80

 so 25 can go into 2,000 80 times

lmao AoPS homework

no clue how the homerun relates but 1 foot is 12 inches so 12 feet in inches is 12x12 because thats how many inches will be in 12 feet so 12x12=144 so your answer is 144

A * B = 84, 
A^2 + B^2 = 193, solve for A, B 
Use substitution and you get:
A = 12  and  B = 7
Sum = 12 + 7 = 19

(a+b)^2=a^2+b^2+2ab

=193+2*84

=361

a+b=19

inverse tan is the same as arctan.   Use atan    

Let the smaller number be n. Then bigger number is n+16, so ratio of bigger to smaller is (n+16)/n, which we can write as 1 + 16/n

This has to equal 1 + 4/7  so we must have 16/n = 4/7

Multiply both sides by 7n to get:  16*7 = 4*n

Divide both sides by 4 to get  16*7/4 = n or n = 28

The larger number is therefore 28 + 16 or 44.

Which of the following points is on a circle if its center is (-13,-12) and a point on the circumference is (-17, -12)?