Question

If the equations $x^2+bx+ca=0$ and $x^2+cx+ab=0$ have a common root, then their other roots are the roots of the equation, $x^2+ax+bc=0.$If you think this is true write 1 otherwise write 0 .

Answer 1

Let the roots of the equation be $\alpha ,\beta$ and $\alpha ,\gamma$ as one root is common.
$\alpha +\beta =-b,\alpha \beta =ca\cdots \left(1\right)$
$\alpha +\gamma =-c,\alpha \gamma =ab.\cdots \left(2\right)$
We are to find the equation whose roots are $\beta$ and $\gamma$ for which we must know the values of $\beta +\gamma ,\beta \gamma .$
$\because x^2+bx+ca=0$ and $x^2+cx+ab=0$ have a common root
$\therefore \frac{x^2}{a(b^2-c^2)}=\frac{x}{a(c-b)}=\frac{1}{(c-b)}\Rightarrow \frac{x^2}{-a(b+c)}=\frac{x}{a}=\frac{1}{1}\Rightarrow a^2=1[-a(b+c)]$
$\Rightarrow a=-(b+c)\Rightarrow a+b+c=0$ is the condition....(3)
Also the common root $x=a\Rightarrow \alpha =a.$ Putting $\alpha =a$ in(1)and (2),we get $\beta =c,\gamma =b$
$\therefore S=\beta +\gamma =b+c=-a,$ by (3)
$P=\beta \gamma =bc.$
Hence the equation whose roots are $\beta$ and $\gamma$ is $x^2-Sx+P=0$ or $x^2+ax+bc=0$