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  4. There are two numbers x making the value of the determinant |1 -2 5 2 x -1 0 4 2 x| equal to 86 . The sum of these two numbers, is

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Q. There are two numbers $x$ making the value of the determinant $\begin{vmatrix}1 & -2 & 5 \\ 2 & x & -1 \\ 0 & 4 & 2 x\end{vmatrix}$ equal to 86 . The sum of these two numbers, is

Determinants

Solution:

$D=\left(2 x^2+4\right)-2(-4 x-20) $
$=2 x^2+8 x+44=86 $
$=2 x^2+8 x-42=0 $
$\Rightarrow x^2+4 x-21=0 $
$\Rightarrow \alpha+\beta=-4$