CPhill

Note that 2  = √2 * √2

So

√2 / 2  =  √2 /   [√2 * √2 ]   =    1 / √2

Starting with 0,  78987  is the 889th palindrome 

[BTW...889 is known as the palindrome's rank ]

Starting with 1, it is the 888th palindrome

This can be verified with the calculator here : 

http://rhyscitlema.com/algorithms/generating-palindromic-numbers/

[ The calculator is about half-way down the page.....there are also other intems of interest related to palindromes on this page ]

A procedure is desribed for finding the nth palindrome, but I don't see one that tells us how to determine the rank of any particular palindrome  [ although I did not look at the page with a fine tooth comb]

Maybe heureka knows of a proceedure to produce this???

Using a proportion we have

125 / 2.5  =  Distance / 4.2      multipl both sides by 4.2

125 (4.2) / 2.5  =  Distance  =  210 m

Note that 74  km  =   74000 m  =  7400000 cm

So...it will be    7400000 / 2000000  =  37/10  cm  = 3.7 cm on the map

-5x^2+20x-17    divide through by - 5

-5 ( x^2 - 4x + 17/5)

Take 1/2 of 4  =  2.......square it = 4.....add it and subtract it

-5 ( x^2 - 4x + 4 + 17/5 - 4)

Factor the first three terms and simplify

-5 [ (x -2)^2  +  17/5 - 20/5 ] 

-5 [ (x - 2)^2  - 3/5]

-5( x - 2)^2  + 3

a = -5   d = -2    and  e = 3

So

a + d + e  =   -5 + (-2) + 3  =   -7 + 3  =  - 4 

49 x^2+ 56x- 64 = 0       divide through by 49

x^2 +  56/49 - 64/49  = 0

x^2 -  8/7 - 64/49  = 0       add 64/49 to both sides

x^2 - 8/7  =   64/49

Take 1/2 of 8/7 = 8/14= 4/7......square it  =  16/49   add it to both sides

x^2 - 8/7 + 16/49  =   64/49 + 16/49

x^2 - 8/7 +  16/49  =  80 / 49         factor the left side

(x - 4/7)^2  = 80/49    

So     a = 1, b = -4/7  and c = 80/49

And   a + b + c  =  1 - 4/7 + 80/49 =   [ 49 - 28 + 80] / 49  =  101 / 49

I should have expanded my answer a tad bit more

We will have a vertical asymptote  ar x  =  -8

As the graph approaches x = -8 from the left , the graph will approach infinity

As the graph approaches x = -8 from the right , the graph will approach  -infinity

And there is a horizontal asymptote at y = 3  which is not crossed by the graph

So.....the range is   ( - infinity, 3) U ( 3 , infinity )

B           C              A            B            B         A           C        B          B       C

       1            →            2            →          3          →         4        →        1

A           D              D            C           C         D           D        A          A       D

Note that  every 4  "transformations"  gets us back to the original orientation

So.......

Dividing  2007 by 4...

We get a remainder of  3/4  which results in the 3rd orientation in the pattern, i.e.,

CBAD

√[ 4x - 3 ]  + 10 / √[ 4x - 3]  = 7

Let √ [ 4x - 3]  = a

So we have

a + 10/a  = 7           multiply through by a

a^2 + 10  = 7a      subtract 7a from both sides

a^2 - 7a + 10  = 0     factor

( a - 5) (a - 2)  = 0

Setting each factor to 0 and solving produces a = 5  or a  = 2

So..... 

√ [ 4x - 3]  =  5      square both sides            

4x - 3  = 25           add 3 to both sides

4x  = 28                divide both sides by 2

x  = 7

or

√ [ 4x - 3] = 2   square bot sides

4x - 3  =   4      add 3 to both sides

4x  = 7           divide both sides by 4

x =  7/4

Hectictar's answer is correct.....I mis-read  the problem....!!!