Question

If each pair in the equations, $x+px+qr=0,x+qx+rp=0,x+rx+pq=0,$ have a common root, find the sum and product of the common roots.

Answer 1

$x^2+px+qr=0$ and $x^2+qx+rp=0$ have common root $a$,

$x^2+qx+rp=0$ and $x^2+rx+pq=0$ have common root $b$,

$x^2+rx+pq=0$ and $x^2+px+qr=0$ have common root $c$,

From the first assumption, a will satisfy both the equations,

$a^2+pa+qr=0$ and $a^2+qa+rp=0$

subtract these two equations,

$(a^2+pa+qr)-(a^2+qa+rp)=0-0$

$=>pa+qr-qa-rp=0$

$=>pa-qa+qr-rp=0$

$=>a(p-q)-r(p-q)=0$

$=>(a-r)(p-q)=0$

$=>a-r=0;p-q=0$

$=>a=r,p=q\dots .\left(i\right)$

The second assumption, b will satisfy both these equations,

$b^2+qb+rp=0$ and $b^2+rb+pq=0$

Subtracting these two,

$=>b(q-r)-p(q-r)=0$

$=>(b-p)(q-r)=0$

$=>b=p,q=r\dots .\left(ii\right)$

The third assumption, c will satisfy both these equations,

$c^2+rc+pq=0$ and $c^2+pc+qr=0$

Subtracting these two,

$=>c(r-p)-q(r-p)=0$

$=>(c-q)(r-p)=0$

$=>c=q,r=p\dots .\left(iii\right)$

Now from equations $\left(i\right)\left(ii\right)$ and $\left(iii\right)$ we see that $p=q,q=r,r=p,$

i.e $p=q=r$ but this will show that all the three equations are same which is not possible because the condition for common root will only be satisfied when the equations are different therefore we discard it,

So we get the common roots $a=r,b=p,c=q$and their product $=abc=pqr$