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  4. If the quadratic equations 3 x2+a x+1=0 and 2 x2+b x+1=0 have a common root, then the value of the expression 5 a b-2 a2-3 b2 is

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Q. If the quadratic equations $3 x^2+a x+1=0$ and $2 x^2+b x+1=0$ have a common root, then the value of the expression $5 a b-2 a^2-3 b^2$ is

Complex Numbers and Quadratic Equations

Solution:

$ 6 x^2+2 a x+2=0$ and $6 x^2+3 b x+3=0$
subtracting $x(2 a-3 b)-1=0 \Rightarrow x=\frac{1}{2 a-3 b}$ (put in any equation)
$\therefore 2 \frac{1}{(2 a-3 b)^2}+\frac{b}{2 a-3 b}+1=0 $
$ 2+b(2 a-3 b)+(2 a-3 b)^2=0$
$ 4 a^2+5 b^2-12 a b+2 a b-3 b^2+2=0$
$ -10 a b+6 b^2+4 a^2+1=0 $
$\Rightarrow 5 a b-3 b^2-2 a^2=1 \Rightarrow B$