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Q.
If equations $2 x^2+3 x+4=0$ and $4 a x^2+5 b x+6 c=0$ (where $a, b, c \in N$ ) have a common root then minimum value of $(a+b+c)$ is equal to
Complex Numbers and Quadratic Equations
Solution:
$\Theta$ Roots of $1^{\text {st }}$ quadratic are imaginary
$\therefore$ Both roots will be common.
$\therefore \frac{4 a }{2}=\frac{5 b }{3}=\frac{6 c }{4}= k $
$a =\frac{ k }{2}, b =\frac{3 k }{5}, c =\frac{2 k }{3}$
For $a , b , c$ to be natural number minimum value of $k$ is LCM of 2,3,5 i.e. 30
$\therefore a =15, b =18, c =20 $
$\therefore a + b + c =53$