Question

Find the nature of the roots of the quadratic equation 3x²-4x+5=0, also change the coefficient of X in the given quadratic equation, such that it has equal roots

Answer 1

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   =     3

                     B   =    -4

                     C   =   -5  

Accordingly,  B2  -  4AC   =

                    16 - (-60) =

                    76

Applying the quadratic formula :

              4 ± √ 76  

  x  =    —————

                   6

Can  √ 76 be simplified ?

Yes!   The prime factorization of  76   is

  2•2•19  

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).

√ 76   =  √ 2•2•19   =

               ±  2 • √ 19  

 √ 19   , rounded to 4 decimal digits, is   4.3589

So now we are looking at:

          x  =  ( 4 ± 2 •  4.359 ) / 6

Two real solutions:

x =(4+√76)/6=(2+√ 19 )/3= 2.120  

or:

x =(4-√76)/6=(2-√ 19 )/3= -0.786

Hope helps....