Question

The sum of the roots of the equation 1/2x^2 - 5/4x - 3 =0 is: 5/2 2 1/2

Answer 1

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