Question

If $\alpha$ ,$\beta$ are the positive roots of a quadratic equation such that $|\alpha -\beta |=1$ and $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{5}{2}$, then the quadratic equation is

• A. $x^2+3x+2=0$
• B. $x^2-3x+2=0$
• C. $x^2-3x-4=0$
• D. $x^2-x+2=0$

Answer 1

Answer: (B)

$x^2-3x+2=0$

$|\alpha -\beta |=1$.
Squaring both side we get, ${\alpha }^2+{\beta }^2-2\alpha \beta =1$.
On solving the other condition $\frac{({\alpha }^2+{\beta }^2)}{\alpha \beta }=\frac{5}{2}$.
By these two equations we get, $\alpha \beta =2$ and $\alpha +\beta =\pm 3$
Therefore, both options A and B are correct.