Question 824722: Please help me with these questions :
1) The equations ax^2 +bx+c =0 and bx^2 +ax+c=0 have the same roots. Show that a+b+c=0 where a is not =b and c is not =0.
2) show that the equation (x-k)^ 2 +2x-k has real roots and hence show that 1
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! 1) If the equations  and  ,
where  and  , have a common root, that root will satisfy
 .
Solving:
If  were a solution of ,  would be zero:
 .
So we know that,
 --> 
We can divide both sides of the equal sign by  , to get
 .
(We know that because ).
Since we have concluded that only  could be a solution to both equations,
substituting  for  must make both equations true.
Substituting into ,
we get
We get the same using the other equation.
The two equations could be  , with roots and  ,
and  , with roots and  .
NOTE:
The only solution that we found that could be a common solution to both equations was .
There must be two more, different solutions, one for each equation.
It cannot be as a double solution to both equations,
because that would require  .
If were the only solution to both equations, the equations would be
 , and
 .
The independent term in both should be
 and  .
That would require .
2) 
So  -->  <--> 
Alternatively,
For real solutions we need the discriminant to be non negative
<-->
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