The number of irrational roots of the equation4xx2+x+3+5xx2−5x+3=−32 is

4xxx+1+3x+5xxx−5+3x=−32x = 0 is not a root. Divide both the numerators anddenominators by x and put x + 3x = y to obtain4y+1+5y−5=−32 ⇒ y=−5,3x + 3x  = - 5 x2+3x=-5 x2+5x+3=0 x=-5±25-122 =-5±132has two irrational roots and x + 3x= 3 x2+3x=3 x2-3x+3=0 x=3±9-122 =3±-32=3±i32has imaginary roots.