Hello,

In order to prove that the first equation (x²+(3a-2)x+a(a-1)=0) has real roots, you have to prove that the discriminant, 5a²-8a+4 is always positive (or equal to 0).

So you have to see when

In order to prove this, find the discriminant : 64-16*5<0

Hence the sign of 5a²-8a+4 is always the same, and its sign is the sign of the leading coefficient, 5.

Thus for all a.

Therefore, the equation has real roots for all a.

41th