This equation has to factor somehow or else you used the quadratic formula.
Let's start with the given equation
1/2 x^2 - 5/4 x - 3 = 0 Multiply the entire equation by 2
2* (1/2x^2 - 5/4 x - 3 = 0) Remove the brackets. The idea is to get rid of the 1/2
2 * 5/4 = 5/2
2*1/2 x^2 - 2(5/4)x - 2*3 = 2*0 Do the multiplication
x^2 - 5/2x - 6 = 0
Now the answer is immediate if you know how to do it. The sum of the roots is - (the middle term's coefficient) which is - (-5/2) which is + 5/2
So the answer is 5/2. That should be the sum of the roots. But we'll carry on just to show that it is right.
The equation factors, not easily, but it does factor.
(x - 4)(x + 1.5) = 0 is the way it factors. You should show this is true.
x^2 + 1.5x - 4x - 6 = 0
x^2 - 5/2 x - 6 = 0 But wait. I did say that the sum of the roots = 5/2. Where did the minus come from?
The answer to that is that we have not yet found the roots. We found the factors.
x1: (x - 4) = 0; x = 4;
x2:(x + 1.5) = 0; x = - 1.5
The sum of the roots = x1 + x2 = 4 - 1.5 = 2.5 which is 5/2
So the sum of the roots = 5/2