CPhill

3x^2+4x-9 = 2x^2-6x+1  rearrange as

x^2 + 10x - 10   =  0

By Vieta

Call the roots a and  b

The sum of the roots =   -10/1   =  -10

So

a +b  =  -10     squsre both sides

a^2 + 2ab  + b^2   =100     (1)

And the product of the roots =  -10/1  = -10

So

ab =  -10     and       2ab   =  -20    (2)

Sub (2)  into (1)

a^2  - 20  + b^2 =  100

a^2 + b^2  =   120   =    the sum of the squares of the roots

Not as hard as it appears

The domain is defined  by the  first  function

Since we can't have a negative under the first radical , x  must be ≥ 2

But notice in the second radical that if x ≥ 2  that this will always result in a positive quantity under this radical

So....the domain is  [ 2, inf )

Simplify as

sqrt [ -3x^2 - x + 6 ] 

The expression under the root  must be  ≥ 0   (we can't take the sq rt  of a  negative)

So

-3x^2 - x +  6  ≥  0

This is a  parabola that opens  downward......all we have to  do is to  find the  roots....the domain will lie  between these roots an also include them....so...

-3x^2  - x  + 6 = 0         multiply through by -1

3x^2 + x - 6 =  0

By the quadratic formula

x =   [ -1 - sqrt [ 1^2 - 4*3* -6 ]/ [ 2 *3 ] =   [ -1 - sqrt ( 73 ) ] / 6

and

x =  [ -1 + sqrt (73) ]  6

So....the domain  is

[    (-1 -sqrt 73) / 6  ,  (-1 + sqrt 73) / 6    ] 

Distance by scooter =  3/4 +1 =  1.75 miles

Time to travel =  1.75 miles * 8 min/mile   =    14 min

If she goes straigt across we can use the P Theorem to  find the distance because it will be the hypotenuse of a  right triangle with legs of 1, 3/4  miles

D =  sqrt  [ 1^2  + (3/4)^2 ]  =   sqrt [ 1 + 9/16 ] = sqrt [  25/16 ]  = 5/4 miles

Time to  travel by foot =  (5/4)miles * 20 min / mile  = 25 min

Faster by scooter  !!!!

Simplifying  f(x)  we have

2x^2 -13x + c

This will be a parabola that turns  upward.....we can let the vertex lie on the line y =1

The x coordinate of the  vertex is  (13 / 4)

And we  want the function  to =  1    ......so......

2(13/4)^2  - 13(13/4)  + c  = 1

2(169) / 16   - 169 / 4  + c  = 1

169 / 8  - 169/ 4 +  c  = 1

-169/ 8  + c  = 1

c =  1 + 169/8

c =  8/8 + 169/8

c = 177 / 8   =   22.125

In what form  ???

x^2 - 8x + n  = 0

If the discriminant is < 0, then we have no real solutions....so

(-8)^2  - 4n < 0

64 < 4n

64/4 < n

n > 16

Any n  from 1 to  10 inclusive will work  !!!!!

We can easiy see this....the disriminant will be smallest when n= 10  →  (8)^2 - 4(10) = 24  but will still be > 0

\($\frac{-5\pm i\sqrt{131}}{6}$ \)

We can  solve for k  in  the  discriminant  thusly

5^2  - 4 (3)(k)  =  131i^2             { i^2  =  -1    and sqrt (-1)  = i }

5^2 - 12k = -131

25 - 12k  = -131

-12k  = -131 - 25

-12k  =  -156

k = -156 / -12  =  13

x(x-3)=1 + 8x - 5

x^2 - 3x  = -4 + 8x

x^2 - 11x  =  -4       complete the square on x

x^2 - 11x + 121/4 =  -4 + 121/4

(x  -  11/2)^2  =  -16/4 + 121/4

(x - 11/2)^2  = 105/4      we only need  one  root  to  find a,b,c

x - (11/2)  =  sqrt (105) / 2

x =  [ 11 + sqrt (105) ] /  2

abc =   (11)(105)(2) =  2310

Some info is missing.....