Let $x$ and $y$ be real numbers which satisfy $$3x^2 + 2xy + 3y^2 = 32. \tag{1}$$ Then we are to find the ranges of the values of $x + y$ and $xy$. First, we set $$a = x + y. \tag{2}$$ By eliminating $y$ from (1) and (2), we obtain the quadratic equation in $x$ $$\mathbf{A}x^2 - \mathbf{B}ax + \mathbf{C}a^2 - 32 = 0.$$ Since $x$ is a real number, we have $$\mathbf{DE} \leqq a \leqq \mathbf{F}.$$ Next, we set $$b = xy. \tag{4}$$ From (1), (2) and (4) we obtain $$b = \frac{\mathbf{G}}{\mathbf{H}}a^2 - \mathbf{I}.$$ Hence from (3) and (5) we have $$\mathbf{JK} \leqq b \leqq \mathbf{L}.$$
Let $x$ and $y$ be real numbers which satisfy
$$3x^2 + 2xy + 3y^2 = 32. \tag{1}$$
Then we are to find the ranges of the values of $x + y$ and $xy$.
First, we set
$$a = x + y. \tag{2}$$
By eliminating $y$ from (1) and (2), we obtain the quadratic equation in $x$
$$\mathbf{A}x^2 - \mathbf{B}ax + \mathbf{C}a^2 - 32 = 0.$$
Since $x$ is a real number, we have
$$\mathbf{DE} \leqq a \leqq \mathbf{F}.$$
Next, we set
$$b = xy. \tag{4}$$
From (1), (2) and (4) we obtain
$$b = \frac{\mathbf{G}}{\mathbf{H}}a^2 - \mathbf{I}.$$
Hence from (3) and (5) we have
$$\mathbf{JK} \leqq b \leqq \mathbf{L}.$$