Question

If $\alpha$ is one of the roots of a quadratic equation $x^2-2px+p=0$, then the other root is equal to

• A. $\frac{\alpha }{2\alpha +1}$
• B. $\frac{\alpha }{2\alpha -1}$
• C. $\frac{2\alpha -1}{\alpha }$
• D. $\frac{2\alpha +1}{\alpha }$

Answer 1

The correct option is B $\frac{\alpha }{2\alpha -1}$

$x^2-2px+p=0$

Let $\alpha$ and $\beta$ be two roots of the equation.
We know,
$\alpha +\beta =2p$ ...(1)
$\alpha \beta =p$ ...(2)
From equation (1) and (2),
$\frac{\alpha +\beta }{2}=\alpha \beta$
$\alpha +\beta =2\alpha \beta$
$\alpha =2\alpha \beta -\beta$
$\alpha =(2\alpha -1)\beta$

$\beta =\frac{\alpha }{2\alpha -1}$
Hence, option $B$ is the correct answer.