CPhill

x^2  - 12x  + k  = 0

We are actually looking for two primes  that sum to 12

7  and 5   seem to fit the bill

So

x^2  -  12x  +   35  = 0  imply  that

(x  - 7) (x - 5)   = 0

And 7 and 5 are the roots.....and  k  = 35

16 gal cover both rooms

And a total of 5 coats of paint are required for both rooms.....so.....each coat must require  16/5 gal

And since we are only painting the large room....only  3 coats *(16/5) gal per coat  =  48/5 gal will be used

So.....the amount left is

16  - 48/5   =

80/5  - 48/5  =

32/5   gal

Rate * Time  =  Distance

So

4 mph * 2 hrs  =  8  miles  to cover

And the toal distance covered in the first 1.5 hrs.  =  3 * 1.5  = 4.5 miles

So....in the last   .5 hrs....you must cover  8  - 4.5 miles  =  3.5 miles

So.....the rate in the last .5 hrs must be

Distance / Time  = Rate =  3.5 / .5  =    7 mph 

The  other root will be

(13  +  sqrt (131))/ 4

In the form, ax^2  + bx  +  c......the product of the roots  =  c / a

So

(13  +  sqrt (131))/ 4  *   (13  -  sqrt (131))/ 4  =   k/2

(169  - 131) / 16  =  k/2

38/16  =  k/2

38/8  =  k

ax^2 + 8x  +  4  = 0

This will only have one solution when the discriminant  = 0

So.....

8^2  -   4a(4)  = 0

64  - 16a  =   0     add 16a to both sdes

64  = 16a      divide both sides by 16

4  =  a

You just have a decimal.....not a "decimal fraction"

If  the  digits repeat the same pattern   [  a cycle of 32 digits], the fraction  expressing this decimal is just  the digits in this pattern the numerator / the same number of 9's in the denominator

For instance.....

.167   (repeating)  is  just   167 / 999

.1477 (repeating)  is just  1477 / 9999  .....etc.

If the digits don't repeat.....then the decimal is irrational and can't be expressed as a fraction

Thanks, X².....!!!

Hairstyles  =  4    [counting "bald"  ]

Eyebrows  = 2

Ears  = 2

Lips  = 2

Shoes  = 2

So......total "personalities"  =  4 * 2 * 2 * 2 * 2  =   64

[  Hey....that's no small potatoes .... ]

For this one

1)  Ignore the decimal

2)  Take the "whole" and subtract the non-repeating part  =  325 -  3   =  322

3) Put this over a number that has the same number of 9's as the repeating part  =  99, followed by the same number of 0's as the non-repeating part  = 0

4)  So we have

322  / 990  =   161  / 495

Verify for yourself that this is equal to the required decimal

I like that method, Melody.......!!!