Multiple choice

Number of possible value(s) of integer '$a$' for which the quadratic equation $x^2+ax+16=0$ has integral roots, is

  1. $4$
  2. $6$
  3. $2$

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For x^2 + ax + 16 = 0 to have integral roots (r1, r2), r1*r2 = 16 and r1+r2 = -a. Possible pairs (r1, r2) are (1,16), (2,8), (4,4), (-1,-16), (-2,-8), (-4,-4). Sums are 17, 10, 8, -17, -10, -8. There are 6 possible values for a.

AI explanation

Let the integral roots be m and n, so by Vieta's formulas their product is m times n = 16 and their sum is m plus n = -a. The integer pairs that multiply to 16 are (1, 16), (-1, -16), (2, 8), (-2, -8), (4, 4), and (-4, -4). These pairs produce six distinct sums for the roots, which are 17, -17, 10, -10, 8, and -8. Consequently, there are 6 possible integer values for a.

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