CPhill

x^2 - 2x + 1   = 0

(x - 1)(x -1)   = 0

(x -1)^2  =  0 

The repeated root is  x = 1

y = -6x^2

y = 5x + c

The parabola's vertex is (0,0)  and it  turns  downward

The slope  of a tangent line  any place on the  parabola  =   -12x

The slope of the line = 5

The x value  on the parabola where the line  =5 can be  found as

-12x =5

x = -5/12

And  y = -6(-5/12)^2   = -25/24

To find c

-25/24 = 5(-5/12) + c

-25/24 = -25/12 + c

25/24  = c  

Here's a graph that shows this : https://www.desmos.com/calculator/urotlebthh

y = -2x^2 + 8x - 15 - 3x^2 - 14x + 25     simplify

 y = -5x^2 - 6x + 10    (1)

x coordinate  of  vertex =    - (-6)  / (2 * -5)  =  -6/10  = -3/5

Put this back into (1) to find the y coordinate of the  vertex

-5(-3/5)^2 - 6(-3/5) + 10  =   59 / 5

Vertex =  (-3/5, 59/5)

( x + 1) (x - 2p)  =  0      (1)

x^2 + (1 -2p)x  -2p =  0

x value of vertex  =   - (1-2p) / (2)  =  (2p - 1)  / 2     {put this  value back into (1) }

[(2p -1)/2 + 1 ]  [(2p -1)/2 - 2p ] = 0      simplify

[p + 1/2 ] [ -p -1/2]  =  0

p+ 1/2 = 0                -p  - 1/2  =  0

p = -1/2                   - p  = 1/2   →  p = -1/2

So   p  = -1/2

3x^2 + 12x - 4 - 2x^2 + 6x + 7 = ___ (x + ___ )^2 + ___

x^2 + 18x + 3     complete the square on x

x^2 + 18x + 81 + 3 - 81

(x + 9)^2  -  78

first blank =  1

second blank  =  9

third blank =  -78

        B

     (11/5)PQ

       Q

       P

     (2/5)PQ

       A              R       (13/8)AR                C 

[ QBC ]    = 

[ BAC ] - [ QAC ]  = 

(18/5)PQ * (21/8)AR  -  (7/5)PQ * (21/8)AR  =     [231/40 ] PQ * AR

[CRQ ]  = 

[QAC ] - [QAR ] 

(7/5)PQ * (21/8)AR  -  (7/5)PQ * AR  =  [91/40 ] PQ*AR

[QBC ] / [CRQ ] =   ( 231/ 40 ) / ( 91/40)  =   231 /  91

60 pi =  pi * r^2  *  (83/360)

60 ( 360  / 83)  = r^2

r =sqrt  [ 60 * 360  / 83  ]   ≈  16.13

n = smallest angle

n + 2d =  largest angle

(n + 2d) - n = 56

2d  =56

d = 28  

n + n + d + n + 2d  =180

3n + 3d  = 180

n + d = 60

n + 28 = 60

n =  32°

No. of diagonals in  a regular polygon =   n ( n - 3)  / 2  where n  = the number of sides

n(n -3)  / 2 =  2n

n^2 - 3n = 4n

n^2 - 7n  = 0

n ( n - 7)  =  0

n - 7  =  0

n = 7

https://web2.0calc.com/questions/polygon_78