AMC 10 Daily Practice Round 1

Complete problem set with solutions and individual problem pages

Problem 21 Easy

Consider the quadratic equation 3x^2 - 5x + a = 0. One root lies in the interval (-2, 0) and the other lies in the interval (1, 3).  Which of the following values of a is NOT possible?

  • A.

    -10

  • B.

    -6

  • C.

    -3

  • D.

    -1

  • E.

    1

Answer:E

The quadratic equation 3x^2 - 5x + a = 0 has one root in the interval (-2, 0) and another root in the interval (1, 3). This is equivalent to the graph of the function f(x) = 3x^2 - 5x + a intersecting the x-axis at points such that one intersection lies within (-2, 0) and the other within (1, 3). Since the graph of f(x) is an upward-opening parabola, to satisfy these conditions, we need: \begin{cases} f(-2) > 0 \\ f(0) < 0 \\ f(1) < 0 \\ f(3) > 0 \end{cases} Calculating each condition, we get: \begin{cases} 22 + a > 0 \\ a < 0 \\ -2 + a < 0 \\ 12 + a > 0 \end{cases} Solving this system of inequalities yields -12 < a < 0. Therefore, the range of possible values for a is (-12, 0). Thus, the final answer is \boxed{D:1}, which is out of the range.

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