SOLUTION: 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 th

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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