CPhill

This is a  30-60-90 right  triangle

The hypotenuse  = 20

One of the legs  =  20 / 2  =  10

And the other  leg  = 10√3

The area  = (1/2) (product of the legs )   = (1/2) (10) ( 10√3)  =  50√3   units^2

2.  x^2 + (2.6)x + 3.6    .....complete the square  on  x

Take (1/2) of 2.6  = 1.3....square it  = 1.69 ....add it and subtract it.......

x^2 + (2.6)x  + 1.69 + 3.6 - 1.69        facto the  first three terms....simplify the rest

(x + 1.3)^2  + 1.91

b = 1.3   c  = 1.91

So

b + c  =  1.3  + 1.91   =  3.21

1.  Set the equations equal

x^2 + 2x + 7  = 6x + b      simplify

x^2 - 4x + (7 - b)   = 0

If there is  a single solution to this, the discriminant must  = 0

Therefore

(-4)^2  - 4 (1) ( 7 - b)   = 0  simplify

16 - 4(7 - b)   = 0

16 - 28 + 4b  = 0

-12 + 4b  = 0

-12  = -4b    divide  both sides  by -4

3  = b

Here's a graph : https://www.desmos.com/calculator/fykdlia1ej....the intersection point is  (2, 15)

y  = √ [ 120 - √x ]      where  y is an  integer

Note that   120 - √x  must be   ≥  0

One way of solving this is with a graph : https://www.desmos.com/calculator/xmrvja05tc

x values     y  values

14400              0

14161              1

13456              2

12480              3

10816              4

 9025               5

 7056               6

 5041               7

 3136               8

 1521               9

  400              10

So....11 values  of x  produce an  integer  output  for  y

So      

Very nice, heureka  !!!!

HAHAHA!!!.... I don't know about that....most of the time....I just make lucky guesses....LOL!!!!

That's OK, Mathhemathh, I almost forgot that, too !!!

If this has rational solutions, then the discriminant must be a perfect square >  0

So we have

11^2  - 4 (5)c

121 - 20c

Note that when c  =2

121 - 20(2)  =

121 - 40 =  81  which is a perfect square

And when  c  = 6

121 - 20(6)  =

121  - 120 =  1  which is also a perfect square

So....the product of these  two values of  c   =  2 * 6    = 12

If  f(6)  = 12

Then  f (3 * x)  = 12

Then  x  = 2

So....this means that   3 f(x)   = 12

Implies that

f ( x)  = 4

So

f (2)  = 4

OK, mathtoo    !!!!!