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Advanced Math / Nonlinear equations in one variable and systems of equations in two variables Difficulty: Medium

$x squared, minus a, x, plus 12, equals 0$

In the equation above, a is a constant and $a, is greater than 0$ . If the equation has two integer solutions, what is a possible value of a ?

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Explanation

The correct answer is either 7, 8, or 13. Since the given equation has two integer solutions, the expression on the left-hand side of this equation can be factored as $open parenthesis, x plus c, close parenthesis, times, open parenthesis, x plus d, close parenthesis$ , where c and d are also integers. The product of c and d must equal the constant term of the original quadratic expression, which is 12. Additionally, the sum of c and d must be a negative number since it’s given that $a, is greater than 0$ , but the sign preceding a in the given equation is negative. The possible pairs of values for c and d that satisfy both of these conditions are $negative 4$ and $negative 3$ , $negative 6$ and $negative 2$ , and $negative 12$ and $negative 1$ . Since the value of $negative a$ is the sum of c and d, the possible values of $negative a$ are $negative 4 plus negative 3, equals negative 7$ , $negative 6 plus negative 2, equals negative 8$ , and $negative 12 plus negative 1, equals negative 13$ . It follows that the possible values of a are 7, 8, and 13. Note that 7, 8, and 13 are examples of ways to enter a correct answer.