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 $\boxed{\textbf{(B)}}$ .

1963 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40
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1963 AHSME Problems/Problem 24

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