Question

(a) 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, $x^2+ax+bc=0$
(b) If the equations $x^2+abx+c=0$ and $x^2+acx+b=0$ have a common root, then establish that their other roots are the roots of the equation $x^2-a(b+c)x+a^{2}bc=0$.

Answer 1

(a) Let the roots of the equation be $\alpha ,\beta$ and $\alpha ,\gamma$ as one root is common,

$\alpha +\beta =-b,\alpha \beta =ca$.........(1)

$\alpha +\gamma =-c,\alpha \gamma =ab$.........(2)

We are to find the equation whose roots are $\beta$ and $\gamma$ for which we must know the values of $\beta +\gamma$ and $\beta \gamma$

$x^2+bx+ca=0$ and $x^2+cx+ab=0$

have a common root

$\therefore$ $\frac{x^2}{a(b^2+c^2)}=\frac{x}{a(c-b)}=\frac{1}{(c-b)}$

or $\frac{x^2}{-a(b+c)}=\frac{x}{a}=\frac{1}{1}\therefore a^2=1[-a(b+c\left)\right]$

or $a=-(b=c)$ or $a+b+c=0$

is the condition. .........(3)

Also the common root $x=a$ i.e. $\alpha =a$.

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 $x^2-ax+bc=0$

(b) Refer part (b). Common root is $1/a$, and other roots are ac and ab.

$\therefore$ $S=a(b+c),P+a^{2}bc$.