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I'm not sure what is going on but post #10 definitely isn't by history 🤣

For the record: Post #5, signed by GA, is not by the real GA.

By the Pythagorean Theorem, the missing length is 8.

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

Hi there!

There are 4 integers for which there is no real solution. They are: 7, 8, 9, and 10. Therefore, the probability is \(\frac{4}{10}=\frac{2}{5}.\)

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

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.....