Let
a1x2+b1x+c1=0‌and‌a2‌x2+b2x+c2=0
be the two quadratic equation since 1 is a common root.
Then,
⇒
a1+b1+c1=a2+b2+c2=0 ⇒
b1=−(a1+c1)‌and‌b2=−(a2+c2) Also, the discriminants of two quadratic equations are equal
Then
⇒
b12−4a1c1=b22−4a2c2 ⇒(a1+c1)2−4a1c1=(a2+c2)2−4a2c2
⇒
(a1−c1)2=(a2−c2)2 ⇒
a1−c1=±(a2−c2) ⇒a1−a1=c1−c2‌or‌a1+a2=c1+c2
Now, the roots of equation,
x=−=a1‌and‌c1 and the roots of equation
2,x==a1and
c2 If
a1=a2=1 be the common roots
Then, other roots are
c1‌and‌c2 ⇒c1−c2=a1−a2=0‌or‌c1+c2=a1+a2 ⇒
c1=c2‌‌‌‌‌‌c1+c2=2 Hence, roots are either equal or their sum is 2.