CPhill

Simplify  as

5x = 4y^2  + 9y  + 13

x = (4/5)y^2 + (9/5)y + 13/5

The y coordinate of the vertex = -(9/5) / (2 * 4/5)  = -9/8

The x coordinate is

(4/5) (-9/8)^2 + (9/5)(-9/8) + 13/5 =   127 / 80

https://web2.0calc.com/questions/right-triangle_44

https://web2.0calc.com/questions/bisectors_25

No prob !!!

Ax ^2 + Bx + 1

2x^2 - 5x + 6      divide this  through by 6

(1/3)x^2 - (5/6) x + 1

A + B   =  (1/3) - (5/6)  =  (2/6) - (5/6)  = -3/6  = -1/2

4x^3 -11x^2 + 2x + 3

Test whether 3/4  is a root

3/4  [  4     -11     2      3   ]

                    3   -6      -3

   ___________________

        4       -8     -4      0

It is  a root

The remaining polynomial is

4x^2 - 8x - 4 = 0

x^2 -2x - 1  =0

x^2  -2x   =  1

x^2 -2x +1  =1 + 1

(x -1)^2  = 2       take both roots

x -1  =sqrt 2                   x - 1 = -sqrt 2

x = 1 + sqrt 2                 x = 1 - sqrt 2

These are the other two roots

Simplify as

3x^2 -kx + 33  = 0

Will have no real solutions when

k^2 - 4*3*33 < 0

k^2  - 396 < 0

k^2 < 396

k < sqrt (396)

k = 19

1  / (64a^3 + 7)  - 7 =  0

1 / (64a^3 + 7)  =  7

1 = 7 (64a^3 + 7)

1 = 448a^3 + 49

-48 = 448a^3

-48 / 448  = a^3

-3/28 = a^3

a = (-3/28)^(1/3)

480 = 2^5 * 3  * 5  = 

2 * 4 * 4 * 3 * 5 * 1 =    6! / ( 2!) =   360

6 * 4 * 4 * 5 * 1 * 1 =  6! / ( 2! * 2!)  =  180

360 + 180  =  

540 possible sequences

Add these

2a  =10

a = 5

5 + 5b  = 250

5b = 245

b = 49