ElectricPavlov

 7x^2 + x - 5 = 6x^2 - 11x - 2   simplifies to :

x^2 + 12x -3  = 0       which has roots     -6 +- sqrt (39)    ( Using Quadratic Formula with   a = 1    b =12   and  c= -3 ) 

                then     a -4 =    -10 + sqrt (39)       and   b-4 =  -10 - sqrt (39)    adding these together :      =    -20 

ANOTHER way :

   we know that    - (a+ b ) = 12 

                   or    a + b = -12          now subtract   -4  and -4   from both sides

                            (a-4) + (b-4) = -12 -4 -4 

                                                = -20 

3x^2 + 17x + 15 = 4x^2 - 8x + 37.

Simplify to 

 x^2 - 25x +22 = 0         Now use Quadratic formula   with    a = 1      b = -25     and   c = 22

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

\(x = {-(-25) \pm \sqrt{(-25)^2-4(1)(22)} \over 2(1)}\)         to find x =   25/2  +-  sqrt(537) / 2     the largest of which is  x = 24.0866  

For roots    x = 5   and x = 0   the quadratic in FACTORED form is 

   (x-5)(x-0) = 0       now expand the L side

x^2 - 5x + 0    = 0 

     x^2 - 5x  +0 = 0 

16 -5 = 11

Here is one way:

log1/2 (11)  means   1/2^x  = 11    LOG both sides

                                 x LOG (1/2) = LOG 11

                                  x = - 3.4595 

Depending on your calculator you may be able to calculate it directly     log_1/2 (11) =   -3.4595    

The PERPINDICULAR line from the origin has a slope of   - 1/m =  -1/2 

  so  this line is  y = - 1/2 x 

   A circle of radius 5 , centered at the origin (  x^2 + y^2 = 5^2 ) 

    Intersects are prpindicular line at   (4.472 , -2.236) 

        NOW....the equation of the line that contains THIS point and has a slope of 2 is 

                     y + 2.236 = 2 ( x - 4.472) 

                   or  y = 2x - 11.18 

This has a y-axis intercept of  - 11.18    and the x intercept is 5.59 

Area of the triangle is 1/2 base * height     

     = 1/2   * 11.18  * 5.59 =  31.25 units^2 

Here is a Desmos graph showing all of this :

1 *-23x    +  14 * x =   - 9x

so  -9 is the coefficient of x 

We are looking for the two points where the equations intersect....where they are equal ...so equate the two circle equations 

x^2 + y^2 -4     =  ( x-5)^2 + (y-8)^2 - 57            expand  ...and then simplify to 

0 = -10x -16y+36    which rearranges to 

y = - 10/16 +36/16 

y = - 5/8 x +9/4      this is the  equation of the line AB......the slope is - 5/8 

Here is the DESMOS graph showing the two circles and the (green) line: