Question

A student notices that the roots of the equation $x^2+bx+a=0$ are each 1 less than the roots of the equation $x^2+ax+b=0$.Then $a+b$ is

• A. possibly any real number
• B. $-2$
• C. $-4$
• D. $-5$

Answer 1

The correct option is C $-4$

Let p, q be roots of first equation $\Rightarrow$p+1,q+1 are the roots of second equation

Applying conditions for 1st equation :

$p+q=b$ and $pq=a$

Applying conditions for 2nd equation :

$p+q+2=-a\Rightarrow a-b=-2$

$pq+(p+q)+1=b$ (product of roots)

$\Rightarrow a-2b=-1$

$\therefore a=-3$ and $b=-1$

$\therefore a+b=-4$