The nature of the roots of a quadratic equation can be determined by the value of its discriminant `D`, which is calculated using the formula `D = b^2 - 4ac`, where 'a', 'b' and 'c' are the coefficients of the quadratic equation written in the standard form `ax^2+bx+c=0`.
In this problem, we are working with three quadratic equations.
- The first equation, 6x^2 -5x -17 = 0, has 'a', 'b' and 'c' values of 6, -5, and -17 respectively. Applying these values to the discriminant formula: D1 = (-5)^2 - 4*6*(-17). After solving this, we find that D1 is positive.
- The second equation, 5x^2 -10x +5 = 0, corresponds to 'a', 'b', and 'c' values of 5, -10, and 5. Again, applying our discriminant formula: D2 = (-10)^2 - 4*5*5. After calculation, we find that D2 is equal to zero.
- The third equation, 9x^2 -7x +4 = 0, gives us 'a', 'b', and 'c' values of 9, -7, and 4. Plugging these into our formula: D3 = (-7)^2 - 4*9*4. Here, D3 is less than zero.
From the calculation of the discriminants, we see that a quadratic equation will not have any real roots if its discriminant is less than zero. Since D3 is negative, the third equation, 9x^2 -7x +4 = 0, has no real roots.