CPhill

x^2  +  10x     =   9

Take 1/2  of 10    = 5

Square this = 25

Add to  both sides

x^2  + 10x +  25  =   9 + 25

x^2 + 10x +  25  =  34       factor the  left  side

( x + 5)^2    =   34

See Melody's  link and  heureka's excellent answer, here  :

Arc Length  =  r (theta in rads)

24.6 =  r  ( 7pi/18)

24.6 *  18  /( 7pi)   = r  ≈  20.135 in

A =  (1/2)r^2  (theta in rads)

A = (1/2) (5^2)  (5pi/12)   =  (125/12)pi  in^2

  Suppose that ABC4 + 200 = ABC9, where A, B, and C are valid digits in base 4 and 9. What is the largest possible value of A + B + C in base 10? 

We have 

A(4)^2  + B(4)   + C  +  200   =   A(9)^2  + B(9)  + C

A( (9^2 - 4^2)   +  B(9-4)   -   200    =    0

65A  +  5B  -  200   =    0

13A  + B  - 40    =    0

13A  + B   =  40

A =  3      B    =  1

As large as C  can  be in  base  4  =    3

A + B + C    =  3 + 1 + 3    =   7

Compute 1 − 3 + 5 − 7 + 9 − 11 + · · · + 97 − 99 + 101   

We  will have  ( 101  + 1)/ 2  =  51 terms

We can write this in a slightly different manner

(101 + 1)  + (-99 - 3) + ( 97 + 5)  + (-95 - 7)  .......   

We will  have 50 pairings which will sum to   0 

The middle  term   will  be unpaired   with any other term  in this summation

And  this term  =   51

So.....the sum =   51

 m is the smallest integer such that 2016m is a perfect square. n is the smallest integer such that 2016/n is a perfect square. What is m/n?    

Factor 2016  =   2^5 * 3^2 * 7

We need to have    m  =   2 * 7   = 14    for   2106m   to be a perfect square

And we need   n to  be  same for 2016/n   to be a perfect square

So.....m/n =  14/14   = 1

There are three cookie bakers, Alice, Bob, and Cindy, in the village. Alice works twice as fast as Bob does, and Bob works twice as fast as Cindy does. During the past 3 hours, the three cookie bakers together made 560 cookies. How many of those cookies were made by Bob?    

Call  the number of cookies that  Cindy can make in one hour = N

Call the number that Bob can make in one hour =  2N

And let Alice make  2 (2N)  = 4N

In  therr hours they can make

3 ( N + 2N + 4N) =   560

3( 7N)  = 560

21N =  560      divide both sides by 21

N =  80/3

So  in 3 hours Bob can  make    3(2N)  = 3(2* 80/3)  =  2 * 80  =     160

OK...glad to help....

8^x                                  9^(x + y)

________  = 64             ________ =   243           

2^(x + y)                          3^(4y)

Write the first as                                  Write  the second as

(2^3)^x                                                     (3^2)^(x + y)                                                                

________  =   64                                   ____________  =    243                           

2^x * 2^y                                                     3^(4y)

2^(3x)                                                        3^(2x) * 3^(2y)

________  =  64                                      _____________   = 243

 2^x * 2^y                                                  3^(2y) * 3^( 2y)                                                   

2^(2x)                                                        3^(2x)

_____  =  64                                           ________  =   243

2^y                                                            3^(2y)

2^(2x - y)  = 2^6                                      3^(2x - 2y)  =  3^5

So

2x - y  = 6   ⇒   -2x + y = -6   (1)

2x - 2y = 5   (2)

add   (1)  and (2)

-y  =  -1

y = 1

2x - 1  = 6

2x  =  7

x =  7/2

2xy  =     2(7/2)(1)   =    7