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lol thats very dumb equations :P

16*4 is 64, and 64 = 4*4*4.

answer is 3.

HOPE THIS HELPED!

can you be a little more descriptive about "No house faces two different streets"?

plug in the values.

-96 = -1x - 10

add ten to both sides.

-86=-1x

x = 86.

HOPE THIS HELPED!

divide by a on both sides to get c + b = d / a

subtract b from both sides.

c = (d / a) - b

HOPE THSI HELPED!!

Thanks, EP........that is an EXCELLENT resource.....!!!!

OK....so we have this

-46 =  6x -13          add 13 to both sides

-33 = 6x                divide both sides by 6

-33 / 6  = x  =  - 11 / 2 =  -5.5

We have this sytem of 6 equations

a + b + c + d + e + f =  2,

32a + 16b + 8c + 4d + 2e + f = 3,

243a + 81b + 27c + 9d + 3e + f = 4,

1024a + 256b + 64c + 16d + 4e + f = 5,

3125a + 625b + 125c + 25d + 5e + f =  6,

7776a + 1296b + 216c + 36d + 6e + f = -113 

This system is very tedious to solve by hand  [ so...I won't ]

We are only interested in the value of   "f"  because this will be the y intercept for g(0)

Using a matrix calculator......f is shown to be 121  = g(0)

Agree with EP......there might be an algebraic way to solve this [ using a Taylor expansion of cos (2x) ] or using  Newton's method......this might be tedious, but i'm not sure.... a graph seems the easiest way to go....

See here

https://www.desmos.com/calculator/jewm1ikbfc

Graphically (per Desmos):   5.967

This one shouldn't be too tough to figure out

16 * 4 = 64

And 4*4*4 = 64

So....3 fours

9

To see why....

Since each is divisible by 3

The first number  can be 3p and the second number can be 3q

So....their product  is  (3p) (3q) = 9pq

So....the product is divisible by 9  ....!!!

Note that we have the following sequence 

1, 8, 15, 22, 29

When the first day is Thursday....the last Saturday is on the 31st

When the first day is Friday......the last Saturday is on 30th

When the first day is Saturday.....the last Saurday is on the 29th

In each case....there will be 5 Saturdays

But when the first day is on Sunday......the next Saturday will fall on the 7th

And we have the following sequence for Saturdays

7 14  21  28

So...only 4 Saturdays when the first day is Sunday...

asdf is correct....

Since the integers do not have to be distinct (different).....we have....

2*2 + 2 + 2  =  8         4*4 + 4 + 4 =  24        6*6 + 6 + 6   =  48           8*8 + 8 + 8  =    80

2*4 + 2 + 4 =  14        4*6 + 4 + 6  = 34        6*8  +6 + 8  =   62           8*10 + 8 + 10 = 98 

........20                                    ........ 44                           ......76                             ......116

........26                                    .........54                           ......90                             ......134

........32                                    .........64                         ......104

........38                                    .........74

........44

10*10 + 10 + 10  =   120          12 * 12 + 12 + 12   = 168            14 * 14 + 14 + 14  = 224

10*12 + 10 +  12 =   142           12*14 +  12 + 14   = 194

                           ..... 164

We have     (7)*(8) / 2  - 1  =   27 different values.....

Huh I can post again, anyways I found another person asking the same question, and the answer there was -19. I'll compare the 2 when I get time.

Thank you !

(x^2 + 3) (x^2 + 1)   = 0

There are no real number solutons to this......

Setting  both factors to 0   we have

x^2 + 3  = 0                    x^2 + 1 =  0

x^2  =  -3                       x^2 = -1

Note that any real number squared is either 0 or positive

So....no real numbers exist that make either equation true

Here is a good explanation Website:

https://betterexplained.com/articles/prehistoric-calculus-discovering-pi/

Bruh Sudhish, have you really stooped down to these levels to look for aops HW answers? Bruh

wait.. you are right -5 does not work :P

oops

5 is extraneous, but that makes -19, so you missed another case, i dont know which 

it was -20... i got every case except for -7

I think he's asking for the proof of pi or why it exists.

first term becomes 2i-1

then subtract, 2i-1-(3-5i)=2i-1-3+5i=-4+7i.

HOPE THIS HELPED!

im just gonna check myself, it becomes x^2-2x+1=-4, so x^2-2x+5.

quadratic formula:

(2 +or- 4i)/2 so

it is 1+2i or 1-2i.

HOPE THIS CORRECTED VERSION HELPED!