Question 1168783: prove that if the sum of the sequence of the roots of the equation ax^2+bx+c=0 is 1 then b^2=2ac+a^2
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
The roots of  are p and q such that
which is from the quadratic formula
Adding p and q has the square root terms cancel out because we have  in both p and q; the only difference is that p has the positive version and q has the negative version. Effectively we're adding  where 
So after those root terms go away, we have
Therefore, the sum of the roots of is 
We're told that the roots add to 1, so we know further that,
Let's plug that into  to see what happens
 Replace 'a' with -b
 Subtract b^2 from both sides
From that last equation, we see that either b = 0 or c = 0.
If b = 0, then a = 0, but that means  isn't quadratic. Also, a = 0 causes division by zero errors in the quadratic formula. So we must make 'a' and b nonzero. This forces c to be zero.
Therefore, the equation is only true if c = 0.
If we pick a nonzero c value such as c = 1, then,
Which is a contradiction.
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