Question

The equation formed by decreasing each root of $a{x}^2+bx+c=0$ by $1$ is $2x^2+8x+2=0$. Then

• A. $a=-b$
• B. $b=-c$
• C. $c=-a$
• D. $b=a+c$

Answer 1

The correct option is B $b=-c$

Let $\alpha$ and $\beta$ be the roots of the original equation
Therefore
$x^2-(\alpha +\beta -2)x+(\alpha \beta +1-(\alpha +\beta \left)\right)$
$=x^2+4x+1=0$ (dividing the whole equation by the coefficient of $x^2$)
Hence by comparing the coefficients we get
$\alpha +\beta =-4+2=-2=-b$
$\alpha \beta +4-2(\alpha +\beta )=\alpha \beta +1-(-2)=1$
$\alpha \beta =-2=c$
Hence
$b=-c$