SOLUTION: 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

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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