<s>[INST] Quadratic equation Suppose that $a+b = 2m_{1}$ and $ab = 4m_{1}^{2}-3m_{2}$. Why is the quadratic equation $$y^{2}-2m_{1}y+(4m_{1}^{2}-3m_{2})=0$$ instead of $$y^{2}+2m_{1}y+(4m_{1}^{2}-3m_{2})=0$$
In other words, why is it $-2m_{1}y$ instead of $2m_{1}y$ in the second term?
 [/INST] $a+b = x$   and $ab = y$ 
$$(a+b)^2 = x^2$$
$$a^2+2ab+b^2 = x^2$$ 
$$a^2+2ab+(x-a)^2 = x^2$$ 
$$a^2+2y+x^2-2xa+a^2 =x^2$$
$$2a^2+2y-2xa =x^2-x^2=0$$ 
$$2(a^2-xa+y) =0$$ 
Equation $(1)$
$$a^2-xa+y =0$$ 
if you put $a=x-b$  then
$$(x-b)^2-x(x-b)+y =0$$ 
$$x^2-2xb+b^2-x^2+xb+y =0$$
Equation $(2)$ 
$$b^2-xb+y =0$$
As you see Equation $(1)$ and Equation $(2)$ are the same Quadratic equation . $a$ and $b$ are roots of the Quadratic equation. As you see $x$ must be minus  $y$ must be plus while writing the quadratic equation.
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