Multiple choice

Suppose that a quadratic polynomial $x^2+bx+1,\,b\in R$, has two zeros which are both real then which one of the following is necessarily true?

  1. $b$ can have infinitely many values
  2. $b$ has a unique value
  3. $b$ has atmost two distinct values
  4. $b$ has atmost four distinct values
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A Correct answer
Explanation

For real roots, the discriminant D = b^2 - 4 >= 0. This means b^2 >= 4, so b >= 2 or b <= -2. This allows for infinitely many values for b.

AI explanation

For the quadratic polynomial x^2 + bx + 1 to have real zeros, the discriminant must be non-negative, meaning b^2 - 4ac is greater than or equal to zero. Substituting a = 1 and c = 1 gives the inequality b^2 - 4 is greater than or equal to 0, so b^2 is greater than or equal to 4. This condition holds true for any real number b satisfying b is greater than or equal to 2 or b is less than or equal to -2, meaning b can have infinitely many values.

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