Question

If the roots of $x^2-bx+c=0$ are each decreased by $2$, then resulting equation is $x^2-2x+1=0$, if and only if

• A. $b=6,c=9$
• B. $b=3,c=5$
• C. $b=2,c=-1$
• D. $b=-4,c=3$

Answer 1

The correct option is A $b=6,c=9$

We know that equation $a{x}^2+bx+c=0$

Then sum of roots $=\frac{-b}{a}$ and product of roots$=\frac{c}{a}$

For the 1st equation,

Let the roots be $\alpha$ and $\beta$

$\alpha +\beta =b$ and $\alpha \beta =c$ ...(1)

As per the given condition,

$(\alpha -2+\beta -2)=2$ ...(2)

$\therefore \alpha +\beta =6$

$\therefore b=6$ ...from (1) & (2)

Also, $(\alpha -2)(\beta -2)=1$ ...(3)

$\therefore \alpha \beta -2(\alpha +\beta )+4=1$

$\therefore c-2\left(6\right)=-3$

$\therefore c=9$