The Other Vieta's Formula

$c$ and $d$ are the roots of the equation $x^2 + ax + b = 0$ , then $c + d = -a$ and $cd = b$

$a$ and $b$ are the roots of the equation $x^2 + cx + d = 0$ , then $a + b = -c$ and $ab = d$

$c + d = -a \rightarrow \space a+c= -d \space ...(1)$

$a + b = -c \rightarrow \space a+ c = -b \space ...(2)$

We can clearly see that $(1)=(2)$ .

Since $b$ and $d$ are not equal to zero, we can easily conclude that $b=d$ .

$cd = b \rightarrow c= 1$ .

$ab = d \rightarrow a=1$ .

$c + d = -a \Rightarrow 1 + d = -1 \rightarrow d=b=-2$ .

Hence, we have $a + b + c + d = 1 + 1 - 2 - 2 = -2$ .