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Q.
If roots of the quadratic equation $b x^2-2 a x+a=0$ are real and distinct, where $a, b \in R$ and $b \neq 0$, then
Complex Numbers and Quadratic Equations
Solution:
Given, $b^2-2 a x+a=0$
As, roots are real and distinct, so $D>0 \Rightarrow 4 a^2-4 a b>0 \Rightarrow a(a-b)>0$ ....(i)
Let $ f(x)=b x^2-2 a x+a$
So, $ f(0)=a$ and $f(1)=(b-a)$
$\Rightarrow f (0) f (1)= a ( b - a )=- a ( a - b )$
$\Rightarrow f (0) f (1)<0 $ (Using equation (i))
So, $ f(x)=0$ will have a root in $(0,1)$.