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(2a-3)(4b-6) = 56.

Not many people on this forum would make sense of this.

You beat me to it Chris  

Note that \({1 \over a - 1} + {1 \over b - 1} = {b-1 \over (a - 1)(b-1)} + {a-1 \over (a - 1)(b-1)} = {a + b - 2 \over ab−a−b+1} = {a + b - 2 \over ab−(a+b)+1} \)

By Vieta's, \(a + b = -{b \over a} = {7 \over 2}\) and \(ab = {c \over a} = {2 \over 2} = 1 \)

Now, just plug these back into the original equation and simplify...

The coefficient of z is \((4 \times 6) + (-7 \times -1) \)

\(144 \Rightarrow 2^4 \times 3^2\)

\(196 \Rightarrow 2^2 \times 7^2\)

\(150 \Rightarrow 5^2 \times 2 \times 3\)

144 has \(5 \times 3 = 15\) factors. 

196 has \(3 \times 3 = 9\) factors, but we need to subtract 3 (1, 2, and 4)

150 has \(2 \times 2 \times 3 = 12 \) factors, but we need to subtract 2 (1 and 2)

So, there are \(15 + 9 + 12 - 3 - 2 = \color{brown}\boxed{31}\) numbers. 

1/ ( a^3 + 7)   - 7 =  -a^3 / ( a^3 + 7)      rearrange as

1 / (a^3 + 7)  + a^3/ (a^3 + 7)  =  7

(1 + a^3)  = 7 ( a^3 + 7)

1 + a^3   7a^3 + 49

-6a^3  = 48

a^3  =  -48 / 6

a^3 =  -8

a  = -2

2.   -3cos ( pi  - 2x)  =  1

Note   

cos (pi -2x)  =  cos (pi) cos (2x)  + sin (pi) * sin (2x)  =  -1cos (2x) + 0* sin (2x)  = -cos (2x)

So

-3 (-cos (2x) )  = 1

3 cos (2x)   = 1

cos (2x)  =  1/3

arccos (1/3)  = x    ≈  .123  rads 

So

2x  =  1.23

x = 1.23 / 2  ≈  .615 rads

So   x =     [ .615  + 2pi * n ] rads      where n is an integer

Also   

x = [ (2pi - .615 )   + 2pi*n] rads   ≈  [   5.668  + 2pi*n ] rads    where n is an integer

1.  Note that  sin ( x - pi)  = sin x cos (pi ) - sin( pi ) cos x  =  sinx * (-1) - 0 * cosx  =  -sin x

So

2sin ( x -pi )  - 1  = 0

2 (-sin x)  - 1 = 0

-2sin x  =  1

sin x  =  -1/2

x =   7pi / 6  + n* 2pi         and  x  = 11pi/6  + n* 2pi

Where n is an  integer

Thanks Chris

1 plays only an instrument

8 play a sport and and instrument

9 play only a sport

5 do neither

                                 Sport       No Sport        Total

Instrument                   8                  1                 9

No Instrument             9                  5                14

                                 _______________________

                                  17                 6                 23

P ( no instrument given  a sport)  =     9  / 17

Thank youuuuu!!!!

19t + 5/t + 2/t = 

[ 19 + 5 + 2 ] / t  =

26 / t

t = 1 , 2 , 13,  26

(5N) / ( N + 7)  =  2

5N  = 2 (N + 7)

5N  = 2N + 14

3N  = 14

N =  14/3

\( (\sqrt{2}+\sqrt{3}+1)^3\)

\(a\sqrt{2} + b\sqrt{3} + c\sqrt{6} + d \)

[ sqrt 2 + sqrt 3  + 1)^3  =

16 + 14 sqrt(2) + 12 sqrt(3) + 6 sqrt(6)                                                   

a= 14

b = 12

c = 6

d = 16

Sum =  48

y =  mx + b

m = slope  =   (5- -11)  /( -4-1)  = -16/5

Using the slope and (1,-11)   we have

y = (-16/5) (x -1)  -11

y = (-16/5)x + 16/5 - 11

y = (-16/5)x  - 39/5        

(16/5)x + y  =   -39/5         ......    multiply through by  (-5/39)

(-16/39)x  - (5/39)y  =  1

And we can  write

x / ( -39/16)  +  y / ( -39/5)  =  1

a =  -39/16

The vertex of \(x^2 + 8x = f(x)\) is \((-4,-16)\).

Now, what number c should we add, the vertex becomes \((-4,2)\)?

Y = your money

B = your brother's money

Initially   Y =  B

Y - 5 = B + 5     →    Y  =  B + 10      (1)

Y + 14  =  2 (B - 14)        (2)

Sub (1)  into (2)

( B  + 10) + 14 =  2 (B - 14)

B + 24  =  2B - 28              rearrange as

24 + 28  = 2B - B

52 = B

Your brother has $52

f(2)  = 2^2 - 2(2)  = 0

Note that, from here on,   every  evaluation of the nested functions will  =   0  because f(0) =  0

So

f(f(f(f(f(f(2))))))  =    0

Let the point  Q =  (x , 6 - 2x)

The distance from  O to A   =  sqrt (200)

The distance  from   O to Q =  sqrt  ( x^2 + (6 -2x)^2 )  = sqrt ( 5x^2 -24x + 36)

The distance from  A to Q =  sqrt [ (x -10)^2  + ( 6-2x - - 10)^2 ]  = sqrt [ ( x -10)^2+ (16 - 2x)^2] =

sqrt [ 5x^2 - 84x + 356  ]

Using the Law of Cosines

200 =  (5x^2 - 24x + 36) + ( 5x^2 - 84x + 356)  -2 sqrt [( 5x^2 -24x + 36)(5x^2 - 84x + 356] cos (60)

200 = (10x^2 - 108x + 392) -2 sqrt [ (5x^2 - 24x + 36) ( 5x^2 - 84x + 356)] cos (60)

This is very messy to solve  (as you might expect !!!)  so I used WolframAlpha as a solver

It gives two values

x ≈ 1.5081    so    y   ≈ -2 (1.5081) + 6 = 2.9838

And

x ≈ 9.0758   so    y ≈  -2 (9.0758) + 6  = -12.1516 

Here's a graph :  

I think that the fact that there is no times sign between the 2 and the bracket makes the 2 a part of the bracket and hence I think that the answer is 1.    I suppose there is some convention somewhere that defines it properly but if normal people don't know the convention that does not help much.

Anyway, it is certainly an indication of why brackets are so important when there is even a hint of ambiguity.  More brackets in this case.