If ax2+bx+c=0 and bx2+cx+a=0 have a common root, then prove that a+b+c=0 and a=b=c - Maths - Quadratic Equations

Question

If ax2+bx+c=0 and bx2+cx+a=0 have a common
root, then prove that a+b+c=0 and a=b=c

Anuradha Sharma · Accepted Answer

Let α is the common root of  ax2+bx+c=0 and bx2+cx+a=0  aα2+bα+c=0 and bα2+cα+a=0α2ab-c2=αbc-a2=1ac-b2From last two we get, α=bc-a2ac-b2so from first and last we get, cb-a22=ab-c2ac-b2

If ax2+bx+c=0 and bx2+cx+a=0 have a common root, then prove that a+b+c=0 and a=b=c.

If ax²+bx+c=0 and bx²+cx+a=0 have a common root, then prove that a+b+c=0 and a=b=c.