CPhill

Nice job, 3Mathketeers !!!!!

From the center of the circle, draw radius 8  to the midpoint of the  chord

Then from  the  center of the  circle draw radius 17  to either endpoint of the  chord

This will will create a right triangle with a hypotenuse of 17 and  one  leg = 8

Half of the  chord length  = the other leg of the  right triangle=  sqrt [ 17^2 - 8^2 ] = sqrt (225) = 15

So.....the length of the  chord = 2 (15) =  30 

The difference of the  roots = 

 sqrt ( 30^2  - 4 * 180) =   sqrt [ 900 - 720 ] =   sqrt [ 180]   =   sqrt [ 36 * 5 ]  =  6 sqrt (5) 

Sum =  1 + 7/8 =   15 / 8

Inverse of a 2 x 2 matrix =

1/ (determinant)  [ d  -b ]

                            [ -c  a]

det  = (2 *-4)  - (-1 * 6) =   -8 + 6 =   -2

So  the inverse matrix  =     (-1/2) [- 4   1  ]  =    [  2    -1/2 ]    

                                                     [  -6    2 ]        [   3    -1 ]      

First blank  "equal"

Second blank "are"

Nice job, tastybananas !!!

Note that both a^2 and  b^2   have AT MOST three digits

But this means that a^2  could have three digits but b^2  could have only one  digit

Since both a and  b are non-zero integers the largest that a could be = 31 so that 31^2   = 961

And the smallest that b could be = 1 so that 1^2 = 1

So  the greatest difference   =

a^2 - b^2  =  961  - 1 =   960

I think that angle C = 45° (rather than 450°)

D is the center of the incircle of ABC

Draw DY and DX

And YD = DX = BX = BY

So YDXB is a square, so angle YDX =  90°

And angle YZX  is an inscribed angle in the  incircle  that has 1/2 the measure of  central angle YDX

So angle YZX = (1/2)(90°) = 45°

B= (-7,4)      C = (3,2)

The midoint of BD =  M =   [ ( -7 + 3) / 2 , (2 + 4) /2] =  (-2, 3)  = the center of the  circle

The  distance   from  B to M =   sqrt [ (-7 - -2)^2 + (4 -3)^2 ] =  sqrt [ 25 + 1] = sqrt (26) = radius of the  circle

So we have

(x - -2)^2  + ( y - 3)^2 = 26

( x + 2)^2  + ( y -3)^2 = 26

x^2 + 4x + 4 + y^2 - 6y + 9 = 26

x^2 + y^2 + 4x - 6y - 13 =  0

x^2 + y^2 - (-4)x - 6y + (-13)  = 0

(p , q, s) = (-4, 6, -13)

Find  \(f(f(f(f(f(4)))))\) 

f(4) = 4 -1 = 3

f (f(4)) = f(3) =  3

f (f(f(4))) = f(3)  = 3

f(f(f(f(4)))) =  f(3) = 3

f(f(f(f(f(4))))) = f(3) = 3