Question

If both $a$ and $b$ belong to the set $\{1,2,3,4\}$, then the number of equations of the form $a{x}^2+bx+1=0$ having real roots is

• A. $10$
• B. $7$
• C. $6$
• D. $12$

Answer 1

The correct option is B $7$

Given, $a$ and $b$ belongs to the set $\{1,2,3,4\}$

Now, determinant of equation $a{x}^2=bx+1=0$ should be positive for its roots to be real i.e, $b^2-4ac\ge 0$ $⟶$ $b^2\ge 4a$

Let's take

$a=1$, then $b^2\ge 4$ which gives values of $b$ as $\{2,3,4\}$

$a=2$, then $b^2\ge 8$ which gives values of $b$ as $\{3,4\}$

$a=3$, then $b^2\ge 12$ which gives values of $b$ as $\left\{4\right\}$

$a=4$, then $b^2\ge 16$ which gives values of $b$ as $\left\{4\right\}$

Therefore total of $7$ equations can be formed with those two variables.