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.
