Question

# If the quadratic equation $${x^2} + ax + b = 0$$ and $${x^2} + bx + a = 0\left( {a \ne b} \right)$$ have a common root, then find the numerical value of $$a + b$$

Solution

## If $$a,x^2+b,x+c_1=0$$and $$a_2x^2+b^2x+c_2=0$$have a common root, then conditions for common root is $$\begin{vmatrix} 1_1 & c_1 \\ a_2& c_2\end{vmatrix}^2=\begin{vmatrix} a_1& b_1\\ a_2& b_2\end{vmatrix}\begin{vmatrix} b_1& c_1 \\ b_2& c_2\end{vmatrix}$$In case ,equation are$$x^2+ax+b=0$$and $$x^2+bx+a=0$$$$\therefore$$ for common root$$\begin{vmatrix} 1& b \\ 1& a\end{vmatrix}^2=\begin{vmatrix} 1& a\\ 1& b\end{vmatrix}\begin{vmatrix} a& b\\ b& a\end{vmatrix}$$$$(a-b)^2=(b-a)(a^2-b^2)$$$$(a-b)^2=-(a-b)(a-b)(a+b)$$$$(a-b)^2=-(a-b)^2(a+b)$$or $$(a-b)^2+(a-b)^2(a+b)=0$$or $$(a-b)^2[1+a+3]=0$$$$(a-b)^2=0$$$$a=b$$but $$\therefore a\neq b$$ as equation become same$$1+a+b+=0$$or $$\boxed{a+b=-1}$$Mathematics

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