Since x = y , need to solve this
(x - 5)^2 + ( x - 15)^2 = 130
x^2 - 10x + 25 + x^2 - 30x + 225 = 130
2x^2 -40x + 120 = 0 divide through by 2
x^2 - 20x + 60 = 0
x^2 - 20x = - 60 complete the square on x
x^2 -20x + 100 = -60 + 100
(x - 10)^2 = 40 take the positive root
x -10 = sqrt (40)
x = 10 + sqrt (40) = 10 + 2sqrt (10) ≈ 16.325
a^2 - b^2 + 2a =
(a + b) (a - b) + 2a = {a + b = -1 }
(-1) (a - b) + 2a =
-a + b + 2a =
a + b
And
a + b = -1
Using the equare of the distance formula
(4 -1)^2 + ( b - 2)^2 = 5^2
9 + (b -2)^2 = 25
(b -2)^2 = 25 - 9
(b - 2)^2 = 16 take both roots
b - 2 = 4 or b - 2 = -4
b = 6 b = -2
Rearrange as
2x^2 -2x = 10 divide through by 2
x^2 - 2x = 5 complete the square on x
x^2 - 2x + 1 = 5 + 1
(x - 1)^2 = 6 take the positive root
x - 1 = sqrt (6)
x = 1 + sqrt (6)
x = (1 + sqrt (6) ) / 1
a + b + c = 8
The denominator cannot = 0
So
y^2 - 5 = 0
y^2 = 5
y = ± 5
The sum of these values = 0
[17 + 10 + 9 + 14 + 16 + 8 + 10 + 10 + sum ] / 11 = 14
[ 94 + sum ] = 14 *11
94 + sum = 154
sum = 154 - 94
sum = 60
The 200th term is 3 + 199 (5) = 998
The sum is
[ 3 + 998 ] * 100 =
100,100
The possible rational roots come from :
All the factors of 45 = ± [ 1, 3 , 5 , 9 , 15, 45 ]
Divided by
All the factors of 2 = ± [ 1, 2]
So we have
± [ 1/2 , 3/2 5/2 , 9/2 , 15/2, 45/2 , 1, 3, 5 , 9, 15 ,45 ]
f(f(x)) = 2(2x + 3) + 3 = 4x + 6 + 3 = 4x + 9
f(f(x)) = 2
4x + 9 = 2
4x = 2 - 9
4x = -7
x = - 7/4
Proof
f (-7/4) = 2 (-7/4) + 3 = -14/4 + 12/4 = -2/4 = -1/2
f ( f(-7/4)) = f(-1/2) = 2 (-1/2) + 3 = 2
The length of the band =
Circumference of a circle with a radius of 5 mm
2pi(5) mm =
10pi mm
Plus
8 x the radius of a pencil (5 mm)
8 x 5 = 40 mm
Total length of band =
[10 pi + 40 ] mm
