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I think the "i" is supposed to be there, MapleTheory

So....

The quantity  under the radical  must be  = -15

So......we can solve this...

7^2  - 4(4)k  = - 15

49  -  16k   =   -15

-16k  =  - 64

k  =  4

thanks so much, guest!

Since you wanted the answer FAST, here is a FAST answer!!!.

Hint: Solve using 4 simultaneous equations:

Answer: a = 24 and b = 20 and c = 6 and d = 14

First.....we can solve

[ 4x + 1 ] / 3  =  1

4x  + 1   =   3

4x  =  2

x  =  2/4  =  1/2

So  ( 1/2, 1)  is on the orginal graph

So..... (1, 1/2)  is on the inverse

So....

  [  f ^-1  (1) ] ^-1  =   [ (1/2) ]^-1  =    1 / (1/2)  =   2

*Since I'm not sure if that "i" before the \(\sqrt{15}\) , I'm going to assume it's not there. If another user knows what to do with the "i" then you should follow their answer.*

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Given the quadratic formula: \(x = {-b \pm \sqrt{b^2-4ak} \over 2a}\) we already know what the variable a and b are 4 and 7, respectively.

Now, we plug in the knowns into the formula, and we get:  \(x = {-7 \pm \sqrt{7^2-4(4)(k)} \over 2(4)}\)

As a result, you get: \(x = {-7 \pm \sqrt{49-16k} \over 8}\)

Now, since we know that the number inside the square root is 15, to solve for k you can do this:

49-16k = 15

-16k = 15-49

-16k = -34

k = -34 / -16

k = 2.125

Therefore, k equals to 2.125

Thanks, ant!!!!

very nice solution, CPhill!

888x   +  889y  =   890

891x   +  892y  =   893

Subtract  the  first equation from the second

3x  + 3y   =  3

x +  y   =  1       ⇒   y =  1 - x

So....using the first equation

888x  +  889 (1 - x)  =  890

888x  +  889  -  889x  =  890

-x  =  1

x  =  - 1

So.....   y  =  1 - (-1)  =    2

So

x  -  y   =    -1  -  2   =    -3

39^2 - 36^2 = 15^2

So, the shorter side = 15 units.

Thanks, hectictar....

Note that  in the "formula"  provided by MIRB, 3, 4, 5 , 6, 9 and 10  all  divide 180,  while ( n - 2) will always be an integer

So...we only need to check the values  n  = 7  and n  = 8

( 7 - 2) * 180 / 7  =   (5/7)*180  ≈  128.57°

(8 - 2) * 180  / 8   =   (3/4)* 180  =  135°

So.....a regular heptagon [ 7 sides ]  will not have integer-valued interior angle measures

Let  f(n)  be the degree measure of an interior angle in the regular polygon.

f(n)  =  [ 180(n - 2) ] / n

f(3)  =  [ 180(3 - 2) ] / 3   =   180 / 3   =   60

f(4)  =  [ 180(4 - 2) ] / 4   =   360 / 4   =   90

f(5)  =  [ 180(5 - 2) ] / 5   =   540 / 5   =   108

f(6)  =  [ 180(6 - 2) ] / 6   =   720 / 6   =   120

f(7)  =  [ 180(7 - 2) ] / 7   =   900 / 7   ≈   128.57

f(8)  =  [ 180(8 - 2) ] / 8   =   1080 / 8   =   135

f(9)  =  [ 180(9 - 2) ] / 9   =   1260 / 9   =   140

So....only  f(7)  is not an integer.  Only one value of  n  results in a regular polygon where the common degree measure of the interior angles is not an integer.

7^3x  = 400    take the log of both sides

log 7 ^3x  =  log 400      and we can write

3x  *  log 7  =  log 400          divide both sides by 3log 7

x  =  log (400)  / [ 3log (7)]  ≈ 1.026  =  1.03

ln  (4)  + ln (3x)  = 2     we can write

ln  ( 4 * 3x)  = 2

ln (12x )   =  2

This says that

e^2  =  12x            divide both sides by 12

e^2  / 12 =   x   ≈  .615  =  .62

x^2  + 7x  +  20  =  0

The sum of the roots  =  -7/1  =  -7

The product of the roots is  20/1  = 20

So.....this implies that in  Ax^2 + Bx + 1  = 0

-B/A  =  -7    ⇒  B/A  = 7 ⇒  B =  7A

And

1/A  = 20  ⇒  A =  1/20

So..... B  = 7(1/20)  = 7/20

So.....  A + B   =    1/20  + 7/20  =  8/20   =  2/5

Happy (belated) birthday, Geno! 

I dont use mac but in the later question related to this chapter( if any) i will still try the command

Thanks for briefly explaining this question cuz i was still a bit lost on this one even after Hectictar explained it 

Happy (belated) Birthday, Geno.....!!!

No, its fine. I got up to 400³= 5^p x 2^q , but never thought of converting 400 to (2⁴ x 5²). Wow! Thanks very much Melody! 

In the form  ax^2  +  bx  + c  =  0

The sum  of  the  roots will be  = -b/a

And the product of the roots will be  = c/a

Sum  of the roots   =   2m

Product of the roots  =  m^2  + n^2

So

2m  =  6/2   ⇒   m  =  3/2

And

m^2  + n^2  =  5/2    ⇒   (9/4) + n^2  =  5/2  ⇒  n^2  = 10/4 - 9/4  =  1/4  ⇒  n  =  1/2

So

m * n   =   (3/2)(1/2)  =  3/4

Notice the pattern

7^3  =  343

7^4  = 2401

7^5  =  16807

7^6  =  117649

7^7 = 823543

7^8  =  5764801

7^9 = 40353607

7^10  = 282475249

So...it appears that  for n ≥ 0   where n is an integer

7^(4n + 3)   has 4 as a tens digit

7^(4n + 4)  has 0  as a tens digit

7^(4n + 5)  has 0 as a tens digit

7^(4n + 6 )  has 4 as a tens digit

So

7^2005   =   7^ ( 4(500) + 5)  ⇒   has  0  as a tens digit

a)  sin θ + cos θ   =   0          square both sides

sin^2 θ  +  2sin θ  cos θ   +  cos^2 θ  =    0

1  +  2sinθcosθ   =  0

2sinθcosθ    =   - 1

sin(2θ)  =  -1

let 2θ  = x

So

sin x  =  - 1        and this occurs at  x =  270°  and at  x  = 630°

So

2θ  =  270°            and       2θ  =    630°

So

θ  =  135°        and   θ  =   315°

We can equate the area as follows

(1/2)base * height =  A  ⇒  65 *  height  =  A      (1)  

120 * (1/2)side  = A   ⇒  60 * side  =  A    (2)

Set (1)  = (2)      and solve for the side

60 * side  =  65 * height

side  = (65/60) * height

side  =  (13/12)* height  =   (13/12)*h

The side  is the hypotenuse of a right triangle with the height and 1/2 base being the legs

So.......Using the Pythagorean Theorem, we have that

sqrt  [  side^2  -  height^2 ]   =  1/2 base

Substitute

sqrt  [  ( (13/12)h)^2  - h^2]  =  1/2 base        

sqrt  [  (169/ 144)h^2 -  (144/144)h^2 ]  =  1/2 base

sqrt [  (169 - 144) h^2    / 144 ]   =  1/2 base

sqrt [   25h^2 / 144 ]  =  1/2 base

(h / 12)  sqrt (25)  =  1/2 base

(5/12)h  =  1/2 base

(5/12)h  =  65

h = (65)(12) / 5  

h  = 13 * 12  =   156  m

So....Area  = 

(1/2)base * height

(1/2)130m * 156m

65m * 156m  =   

10140 m^2

Thank you very much. I understand your explanation and is quite acceptable to me.

I shall give you an accurate answer, even though some people don't like it, because there are "formal methods" of solving such modular equations, using "Euler's totient function" and "Chinese Remainder Theorem" and other formal methods etc.

The method I use, which is considered "ineffcient!!" is a simple heuritic method of trial and error basically, and finds the answer very quickly. It begins by taking the largest modulus, 111,727 in this case, and searches all numbers beginning with 1 as follows: it starts with 1 X 111,727 + 105,788=217,515. Then it takes this number and tests it on the 2nd modular equation by subtracting the remainder of 4,739 and dividing by its modulus of 11,527 and see if it divides it EVENLY. It it does, then it has found the answer. If it doesn't, then it tries 2, 3, 4, 5......etc. until it finds the A, B, C, D. This is perfectly acceptable in the Math World, but is considered "inefficient", especially when the numbers get to be very large.

So, based on the above explanation, here is my solution to your modular equations and good luck to you !!.

A * 427 + 277=B *1,427 + 1,125 =C * 11,527 + 4,739=D * 111,727 + 105,788. Using simple iteration, we have:

A = 2,602, B = 778, C= 96, D = 9. Therefore:

N=[2,602 x 427] + 277 =1,111,331- smallest positive integer.

Since the LCM of {427, 1,427, 11,527, 111,727}=112,105,840,448,063, therefore:

N=112,105,840,448,063n + 1,111,331, where n=0, 1, 2, 3.....etc.