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}$$
2x^2-px+q=0
x=1/2, -2
what is p-q?
We have that
2(1/2)^2 -(1/2)p + q = 0 → (1/2) - (1/2)p + q = 0 (1)
And .... 2(-2)^2 - (-2)p + q = 0 → 8 + 2p + q = 0 (2)
And subtracting (1) from (2) we have 15/2 + (5/2)p = 0 → 15 + 5p =0 → p = -3
And using (2), we have 8 - 6 + q = 0 → 2 + q = 0 → = -2
So, our function is ..y = 2x^2 -3x - 2
So....p - q = -3 - (-2) = -1
x=1/2, -2 what is p-q?
$$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}$$