For $\alpha, \beta \in \mathbb{R}$, suppose the system of linear equations $$ \begin{aligned} & x-y+z=5 \\ & 2 x+2 y+\alpha z=8 \\ & 3 x-y+4 z=\beta \end{aligned} $$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of :

Question

For $\alpha, \beta \in \mathbb{R}$, suppose the system of linear equations

$$ \begin{aligned} & x-y+z=5 \\ & 2 x+2 y+\alpha z=8 \\ & 3 x-y+4 z=\beta \end{aligned} $$

has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of :

$x^2+18 x+56=0$

$x^2-10 x+16=0$

$x^2+14 x+24=0$

$x^2-18 x+56=0$

Solution

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