Question

Middel term explating of this equation 8x^2-21x-72=0

Answer 1

8x^2+-21x-72=0
Solving    8x2-21x-72 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  x  , the solution for   Ax2+Bx+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B2-4AC
  x =   ————————
                      2A 

  In our case,  A   =     8
                      B   =   -21
                      C   =  -72 

Accordingly,  B2  -  4AC   =
                     441 - (-2304) =
                     2745

Applying the quadratic formula :

               21 ± √ 2745 
   x  =    ——————
                      16

Can  √ 2745 be simplified ?

Yes!   The prime factorization of  2745   is
   3•3•5•61  
To be able to remove something from under the radical, there have to be  2  instances of it (because we are taking a square i.e. second root).

√ 2745   =  √ 3•3•5•61   =
                ±  3 • √ 305 

  √ 305   , rounded to 4 decimal digits, is  17.4642
 So now we are looking at:
           x  =  ( 21 ± 3 •  17.464 ) / 16

Two real solutions:

 x =(21+√2745)/16=(21+3√ 305 )/16= 4.587 

or:

 x =(21-√2745)/16=(21-3√ 305 )/16= -1.962 

Two solutions were found :

 x =(21-√2745)/16=(21-3√ 305 )/16= -1.962

 x =(21+√2745)/16=(21+3√ 305 )/16= 4.587