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  4. If the expression x 2+ Kxy +7 x +4 y +10 resolved into two linear factors then the sum of all integral values of K is equal to

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Q. If the expression $x ^2+ Kxy +7 x +4 y +10$ resolved into two linear factors then the sum of all integral values of $K$ is equal to

Complex Numbers and Quadratic Equations

Solution:

$ a=1, b=0, c=10, f=2, g=\frac{7}{2}, h=\frac{K}{2} $
$a b c+2 f g h-a^2-b^2-c^2=0 $
$\Rightarrow 2 \times 2 \times \frac{7}{2} \times \frac{K}{2}-4-10 \times \frac{K^2}{4}=0$
$\Rightarrow 5 K^2-14 K+8=0 $
$\Rightarrow (K-2)(5 K-4)=0 $
$K=2 $