Question 766741
A quadratic equation is {{{ax^2 + bx + c}}}.  Remember that a quadratic equation has real roots if the discriminant is greater than or equal to 0.  The discriminant is {{{b^2 - 4ac}}}.<P>
For the given equation b = 3k - 2 and {{{b^2 = 9k^2 - 12k +4}}}<P>
a = 1 and c = k(k-1) = {{{k^2 - k}}}<P>
The discriminant is therefore (((9k^2 - 12k +4 - 4k^2 +4k = 5k^2 -8k +4}}}<P>
That is a quadratic equation with a=5, b = -8 and c =4.  It has to be greater than or equal to 0 in order for the original equation to have real roots for all values of k.<P>
The discriminant of this equation is 64-80 = -16.  That means this equation has no real roots.  In other words, it's a parabola that never crosses the x-axis (it has no real zeroes.)  Since a is positive, we know the parabola opens up.  Since it opens up, and it never crosses the x-axis, its values are all positive. <P>
 Since the discriminant of the original equation is itself a parabola with only positive values, the original equation has a positive discriminant, and thus only real roots.