Question

If x^2+bx+c=0,x^2+cx+b=0(b is not equal to c)have a common root then b+c=

Answer 1

Let $x_1$ and $x_2$ be the roots of the equation $x^2+bx+c=0.$ Then,

• $x_1+x_2=-b\quad\quad\dots(1)$

• $x_1x_2=c\quad\quad\dots(2)$

Let $x_1$ be the common root of both the equations. Then we get,

$\longrightarrow (x_1)^2+bx_1+c=(x_1)^2+cx_1+b$

[Both hand sides are individually equal to zero.]

$\longrightarrow bx_1+c=cx_1+b$

$\longrightarrow(b-c)x_1=b-c$

Since $b\neq c,$

$\longrightarrow x_1=1$

So from (2),

$\longrightarrow x_2=c$

And from (1),

$\longrightarrow 1+c=-b$

$\longrightarrow\underline{\underline{b+c=-1}}$

Hence -1 is the answer.