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  4. The product of the roots of the equation√x2-4 x+3+√x2-7 x+12=3 √x-3 is

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Q. The product of the roots of the equation$\sqrt{x^2-4 x+3}+\sqrt{x^2-7 x+12}=3 \sqrt{x-3}$ is

Complex Numbers and Quadratic Equations

Solution:

$\sqrt{x^2-4 x+3}+\sqrt{x^2-7 x+12}=3 \sqrt{x-3}$
is defined for $x=3$ or $x \geq 4$.
Write (1) as
$\sqrt{x-3}[\sqrt{x-1}+\sqrt{x-4}]=3 \sqrt{x-3} $
$\Rightarrow x=3 \text { or } \sqrt{x-1}+\sqrt{x-4}=3 $
$\Rightarrow x=3 \text { or } \sqrt{x-1}=3-\sqrt{x-4} $
$\Rightarrow x=3 \text { or } x-1=9-6 \sqrt{x-4}+x-4$
$\Rightarrow x=3 \text { or } 6 \sqrt{x-4}=6$
$ \Rightarrow x=3 \text { or } x=5 $
$\text { Thus, product of roots }=15 $.