CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


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

Suggest Corrections
thumbs-up
 
0


similar_icon
Similar questions
View More


similar_icon
People also searched for
View More



footer-image