If the equations $x^2+a x+b=0$ and $x^2+b x+a=0(a
eq b)$ have a common root, then $a+b$ is equal to

Question

If the equations $x^2+a x+b=0$ and $x^2+b x+a=0(a 
eq b)$ have a common root, then $a+b$ is equal to

Aicharya · Accepted Answer

Let $\alpha$ be the common root, then $ \begin{array}{rlrl} & \alpha^2+a \alpha+b & =0 \ 	ext { and } & \alpha^2+b \alpha+a & =0 \ 	herefore & & \frac{\alpha^2}{a^2-b^2} & =\frac{\alpha}{b-a}=\frac{1}{b-a} \ \Rightarrow & & \frac{\alpha}{b-a} & =\frac{1}{b-a} \Rightarrow \alpha=1 \ 	ext { Now, } \quad & \frac{\alpha^2}{a^2-b^2} & =\frac{\alpha}{b-a} \ \Rightarrow & \frac{1}{(a+b)(a-b)} & =\frac{1}{-(a-b)} \ \Rightarrow & a+b & =-1 \end{array} $

If the equations $x^2+a x+b=0$ and $x^2+b x+a=0(a \neq b)$ have a common root, then $a+b$ is equal to

Solution