Question

if ax2-bx+c and bx2-ax+c have a common factor x-1 then show that c=0 and a=b

Answer 1

x-1=0
x=1
As they have a common factor they both leave a remainder zero when divided by x-1
a(1)^2-b(1)+c=0
a-b+c=0.........(1)

b(1)^2-a(1)+c=0
b-a+c=0......(2)
(1)+(2)
a-b+c=0
-a+b+c=0
a and b get cancelled
2c=0
c=0

Substituting c in eq 1
a-b+0=0
a=b
Hence proved


Hope it helps!!!

Answer 2

Let given quadratic equations, be
f(x)=ax^2-bx+c=0
g(x)=bx^2-ax+c=0
and (x-1) is a factor of both the polynomial.
Then zero of both the polynomial is 1.
Now we have,
f(1)=0
a(1)^2-b(1)+c=0

a-b+c=0....................................................1

g(1)=0
b(1)^2-a(1)+c=0

b-a+c=0....................................................2
By eq1 and eq2 we have,
a-b+c=b-a+c

2a=2b

a=b.............................................................3
putting eq3 in eq1 we get,
b-b+c=0

c=0
Hence showed that ,
a=b and c=0