CPhill

t^2 -9t - 36 + t^2 - 5t + 14        simplify  as

2t^2  - 14t  - 22

The t that produces a min  is the  x coordinate of the  vertex of this parabola given by

- (-14)  / (2 * 2)  =   14/4  =  7/2  =   3.5

 1.2n-4.4<5.2+1.1n        multiply through by 10 to clear the decimals

12n - 44 < 52  +11n

12n - 11n  < 52 + 44

n < 96

The sum of the positive integers from  1  to 96 inclusive is

[96] [ 97] /  2  =

48 * 97  =  

4656

A non-graphical method

The center of the circle  = (0,0)

The equation of the line in standard form  is

2y =  3x + 7

3x - 2y + 7  =  0                 using (0,0) for (x,y).....the distance  from  the circle's center  to the  line is  given  by

l  3(0)  - 2(0)  + 7 l                    7

_________________  =       _________        (1)

     sqrt  [ 3^2 + 2^2]              sqrt (13)

We can  form  a right  triangle  with a hypotenuse of 5, one  leg  = (1)   and the  other leg 1/2 the  chord length

So......the length of the  whole  chord is

2sqrt   [ 5^2 - (7/sqrt(13))^2 ]  =

2sqrt [  25  -  49/13 ]  =

2sqrt [ 276/13 ] ≈   9.215 =  9.22 (rounded to the  nearest hundreth )

Note     1/a^2  + 1/b^2 =      [a^2 + b^2 ] / [ ab ] ^2

The sum of the roots =   11/5  =  a + b

Square both sides

121/25 =  a^2 + 2ab + b^2

The product of the roots =    4/5 = ab     so.....2ab =  8/5        and [ab]^2 = 16/25

So 

121/25 = a^2 + 2(4/5) + b^2

121/25 = a^2 + 8/5 + b^2

121/25  - 8/5 = a^2 + b^2

121/25 - 8/5 = a^2 + b^2

121/25 - 40/25 = a^2 + b^2

81/25 =  a^2 + b^2

So

1/a^2  + 1/b^2 =   [ 81/25] / [ 16/25 ] =    81 / 16

I think you might mean  this  ???

(a + 6i) ( 5 - 3i)

5a + 30i - 3ai  - 18i^2

5a + 30i - 3ai + 18

a =  10

Because

5 (10)  + 30i  -3(10)i + 18  =

50 + 30i - 30i + 18  =

68

A graph might work best :

AB ≈   9.22 units

196  =  1 + 6 + 9  =  16

sqrt (196) = 14     

(1 + 4)^2  =  25

The 10th term  =   

18 * r^6 =  162

r^6  = 162 / 18  =  9                take the 9th root

r=  9^(1/6)

The 7th term is

18 * r^3  =

18 * (9^(1/6))^3  = 

18  *  (9)^(1/2)  =

18 * 3  =  

54

We have a lower power polynomial over a higher power polynomial

This produces a  horizontal asymptote of     y =  0

This  is the bottom of the range

To discover the max use some Calculus

Take the first derivative of the function and set =  0

0 - 2 * (4 + 6x)

____________  =   0

(2 + 4x + 3x^2)

This simplifies to

4 + 6x =  0

4 =  -6x

x =  -4/6 =  -2/3

The max occurs at  x  = -2/3

The max is           2 /  ( 2 + 4(-2/3) + 3(-2/3)^2 )  =    2 / [ 2 -8/3 + 4/3 ] =  2 / [  2/3] =  3

The range is   (0, 3 ]

a + b =   0 + 3  =  3

y^2 - x + 5y - 25 =  x^2 - 6x + 1  rewrite as

y^2 + 5y - x^2 + 5x  - 26 =  0 

-1x^2 + 0xy + 1y^2 + 5x + 5y  - 26 = 0

General (Standard Form) Equation Of A Conic Section

Ax^2+Bxy+Cy^2+Dx+Ey+F=0,where A,B,C,D,E,F are constants

From the standard equation, it is easy to determine the conic type eg

B^2−4AC<0 , if a conic exists, then it is a circle or ellipse

B^2−4AC=0, if a conic exists, then it is a parabola

B^2−4AC>0 , if a conic exists, it is a hyperbola

We have  :

0^2 - 4 (-1) (1)   =   4  >  0

This is a hyperbola that opens upward / downward