DS Manhattan flashcard confusion

[MrCleantek]

I was referring to Manhattan flashcards and saw this question
--------
Is the statement sufficient?
What are the solutions to
x^2 - 10x + b = 0?
1) The sum of the roots is 10.

The explanation given is this:
Answer: Insufficient
Since the middle term of the quadratic expression is −10x, we
know the factored form would take the form (x - a)(x - b),
where a + b = 10. Thus we already knew the sum of the
roots is equal to 10 before statement (1), so it is not enough
information.
--------------
But is this not the solution? (x-1)(x-9)
(x-a)(x-b)
x^2-x(a+b)+ab
From x^2 - 10x + b = 0, ab=b. Hence a=1. Therefore b=9

[Anurag@Gurome]

But is this not the solution? (x-1)(x-9)
(x-a)(x-b)
x^2-x(a+b)+ab
From x^2 - 10x + b = 0, ab=b. Hence a=1. Therefore b=9

You are assuming a pair of values for a and b.
In other words, you are assuming b = 10

Consider the following examples for different values of b,

• 1. (x² - 10x + 9) = (x - 1)(x - 9)
  2. (x² - 10x + 16) = (x - 2)(x - 8)
  3. (x² - 10x + 21) = (x - 3)(x - 7)

So we have different solution set for the equation for different values of b.

I guess you are mixing up with the 'b' in the given equation (x² - 10x + b) = 0 and 'b' in the expression (x - a)(x - b). They are different. The solution should not have used same constants.

[MrCleantek]

Got it. Thanks. Yeah the usual trap, assumption