What is the solution of the equation x2 + 12x + 23 = 0?

−

sqrt of -13 =−9.606

Step-by-step explanation: Root plot for :  y = x2+12x+23

Axis of Symmetry (dashed)  {x}={-6.00}  

Vertex at  {x,y} = {-6.00,-13.00}  

x -Intercepts (Roots) :

Root 1 at  {x,y} = {-9.61, 0.00}  

Root 2 at  {x,y} = {-2.39, 0.00}  

Solving    x2+12x+23 = 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   =     1

                     B   =    12

                     C   =   23

Accordingly,  B2  -  4AC   =

                    144 - 92 =

                    52

Applying the quadratic formula :

              -12 ± √ 52

  x  =    ——————

                     2

Can  √ 52 be simplified ?

Yes!   The prime factorization of  52   is

  2•2•13  

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).

√ 52   =  √ 2•2•13   =

               ±  2 • √ 13

 √ 13   , rounded to 4 decimal digits, is   3.6056

So now we are looking at:

          x  =  ( -12 ± 2 •  3.606 ) / 2

Two real solutions:

x =(-12+√52)/2=-6+√ 13 = -2.394

or:

x =(-12-√52)/2=-6-√ 13 = -9.606