Question

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?

• A. $b$ can have infinitely many values
• B. $b$ has a unique value
• C. $b$ has atmost two distinct values
• D. $b$ has atmost four distinct values

Answer 1

The correct option is A $b$ can have infinitely many values

$x^2+bx+1:b\in R$

Comparing it with general form of quadratic equation $a{x}^2+bx+c=0$,

we have $a=1,b=b,c=1$

Given that roots are real, the discriminant will have to be non-negative.
$⟹D=b^2-4ac\ge 0$
$b^2-4\ge 0$
$(b-2)(b+2)\ge 0$
$(b+2)\le 0$ or $(b-2)\ge 0$
$b\le -2$ or $b\ge 2$
$b$ has infinitely many values