quadratic formula

 x=1/2, -2      what is p-q? 

$$x_1 = \frac{1}{2} \ and \ x_2=-2$$

$$2x^2-px+q=0 \quad | \quad :2 \\ x^2\underbrace{-\frac{p}{2}}_{P}x\underbrace{+\frac{q}{2}}_{Q}=0\\ x^2+ Px+Q=0 \\\\ \[[Vieta\]]\ : -(x_1+x_2) = P = -\frac{p}{2} \ and \ x_1*x_2 = Q = \frac{q}{2} \\\\ \small{\text{ $ p-q = -2P-2Q = (-2)[ -(x_1+x_2) ] - 2x_1*x_2 = 2(x_1+x_2-x_1*x_2) =2[\frac{1}{2}-2-\frac{1}{2}*(-2)]=2(\frac{1}{2}-2+1)=-1 $ }} \\ \textcolor[rgb]{1,0,0}{p-q=-1}$$

2x^2-px+q=0

x=1/2, -2

what is p-q?

We have that

2(1/2)^2 -(1/2)p + q = 0  →  (1/2) - (1/2)p + q = 0       (1) 

And .... 2(-2)^2 - (-2)p + q  = 0     →     8 + 2p  + q  = 0   (2)

And subtracting (1) from (2) we have  15/2 + (5/2)p = 0    →  15 + 5p =0  → p = -3

And using (2), we have 8 - 6 + q = 0   → 2 + q  = 0    →  = -2

So, our function is ..y =  2x^2 -3x - 2

So....p - q  = -3 - (-2)  = -1