Question

If 'a' and 'b' can take values 1, 2, 3, 4, then the number of equations of the form $a{x}^2+bx+1=0$ having real roots is ____.

• A. 10
• B. 6
• C. 7
• D. 12

Answer 1

The correct option is C 7

$a{x}^2+bx+1=0$ has real roots only if D ≥ 0 i.e., $b^2-4ac\ge 0$.

Given that 'a' and 'b' can have any value among 1, 2, 3 and 4. Out of all the possible combinations, only 7 combinations which are (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) and (4,4) satisfy the above condition.

For example:
(a,b) = (1,2)
$\Rightarrow D=b^2-4ac=(2{)}^2-4(1\left)\right(1)=0\ge 0$

(a,b) = (3,4)
$\Rightarrow D=b^2-4ac=(4{)}^2-4(3\left)\right(1)=4\ge 0$

This can be verified for the other values of 'a' and 'b' also.
$\therefore$ 7 equations can have real roots.