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  4. If f (x) is a quadratic expression such that f (1)+f (2)=0, and -1 is a root of f (x)=0, then the other root of f (x)=0 is :

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Q. If $f (x)$ is a quadratic expression such that $f (1)+f (2)=0$, and $-1$ is a root of $ f (x)=0$, then the other root of $f (x)=0$ is :

JEE MainJEE Main 2018Complex Numbers and Quadratic Equations

Solution:

Given one root of $f(x)$ is $-1$.
Then, $f(x)=a(x+1)(x-\alpha), \alpha$ is other root of quadratic equation.
So$f(1)+f(2)=0 $
$\Rightarrow 2 a(1-\alpha)+3 a(2-\alpha)=0 $
$\Rightarrow 2-2 \alpha+6-3 \alpha=0 $
$\Rightarrow \alpha=8 / 5$