Multiple choice

In the quadratic equation a2 - 4a + |M| = 0, where M is an integer, for which values of M does the equation have two distinct real roots?

  1. (-4, 4)

  2. (-2, 2]

  3. (-2, 2)

  4. (-1, 3]

  5. (-3, 2)
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A Correct answer
Explanation

For a quadratic equation to have two distinct real roots, the discriminant D must be > 0. D = (-4)^2 - 4(1)(|M|) = 16 - 4|M|. 16 - 4|M| > 0 => 4|M| < 16 => |M| < 4. This means -4 < M < 4.

AI explanation

For the quadratic equation a^2 - 4a + |M| = 0 to have two distinct real roots, the discriminant must be strictly greater than zero, so (-4)^2 - 4(1)(|M|) > 0. This simplifies to 16 - 4|M| > 0, which means |M| < 4. Since M is an integer, the possible values for M are -3, -2, -1, 0, 1, 2, and 3, giving the set (-4, 4) in interval notation.

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