Multiple choice

lf the difference between the roots of the equation $x^{ 2 }+ax+1=0$ is less than $\sqrt{5}$, then the set of possible values of $a$ is

  1. $(-3,3)$
  2. $(-3, \infty)$
  3. $(3, \infty)$
  4. $(-\infty, -3)$
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A Correct answer
Explanation

For x^2 + ax + 1 = 0, roots are r1, r2. |r1 - r2| = sqrt(D)/a_coeff = sqrt(a^2 - 4). We need sqrt(a^2 - 4) < sqrt(5), so a^2 - 4 < 5, which means a^2 < 9. Thus, -3 < a < 3.

AI explanation

The difference between the roots of the quadratic equation x squared plus ax plus 1 equals 0 is given by the square root of the discriminant, which equals the square root of (a squared minus 4). We are given that this difference is less than the square root of 5, so the square root of (a squared minus 4) is less than the square root of 5. Squaring both sides yields a squared minus 4 is less than 5, which simplifies to a squared is less than 9. Therefore, the possible values of a are in the interval (negative 3, 3).

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