#3x^2 - 5x +7=0# is a quadratic equation. This can be solved in 3 ways: factorising, using the quadratic formula or by completing the square. It's not obvious how to factorise #3x^2 - 5x +7=0# so we'll use the quadratic formula:
#x= (-b+-sqrt(b^2 - 4ac))/(2a)#
a is the coefficient of #x^2#, in this case 3.
b is the coefficient of #x#, in this case -5.
c is the constant, in this case 7.
Putting these values into the quadratic equation:
#x= (-(-5)+-sqrt((-5)^2 - 4(3)(7)))/(2(3))#
#x= (5+-sqrt(25 - 84))/(6)#
#x= (5+-sqrt(- 59))/(6)#
#x= (5+-isqrt(59))/(6)#
So the roots are:
#x= (5+isqrt(59))/(6)#
and,
#x= (5-isqrt(59))/(6)#
#sqrt(-59)# is an imaginary number, equal to #sqrt(-1)sqrt(59)#, which equals #i sqrt(59)#. The symbol for the square root of -1 is i. Thus the roots of this equation have a real part (the 5/6) and an imaginary part (the #+-(isqrt(59))/(6)#). A number made up of a real and imaginary part is called a complex number. Hence there are no real roots to this quadratic, which means it does not cross the x-axis.
