1963 AHSME Problems/Problem 24

Problem

Consider equations of the form $x^2 + bx + c = 0$. How many such equations have real roots and have coefficients $b$ and $c$ selected from the set of integers $\{1,2,3, 4, 5,6\}$?

$\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 19 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 16$

Solution

The discriminant of the quadratic is $b^2 - 4c$. Since the quadratic has real roots, \[b^2 - 4c \ge 0\] \[b^2 \ge 4c\] If $b = 6$, then $c$ can be from $1$ to $6$. If $b = 5$, then $c$ can also be from $1$ to $6$. If $b=4$, then $c$ can be from $1$ to $4$. If $b=3$, then $c$ can be $1$ or $2$. If $b=2$, then $c$ can only be $1$. If $b = 1$, no values of $c$ in the set would work.

Thus, there are a total of $19$ equations that work. The answer is $\boxed{\textbf{(B)}}$.

See Also

1963 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
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