Multiple choice

The coefficients b and c of the equation x2 + bx + c = 0 are determined by throwing an ordinary die. The probability that the equation has equal roots is

  1. 1/18

  2. 13/18

  3. 5/18

  4. 1/9
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A Correct answer
Explanation

For equal roots, discriminant b^2 - 4c = 0, so b^2 = 4c. Possible values for b and c (1 to 6): If b=1, c=1/4 (no). If b=2, c=1 (yes). If b=3, c=9/4 (no). If b=4, c=4 (yes). If b=5, c=25/4 (no). If b=6, c=9 (no). There are 2 successful outcomes out of 36 total combinations (6*6). Probability = 2/36 = 1/18.

AI explanation

For the quadratic equation x squared + bx + c = 0 to have equal roots, its discriminant must equal zero. Using the discriminant formula b squared minus 4ac with a = 1 and c = c, we get b squared minus 4c = 0, or b squared equals 4c. Since b and c are determined by rolling a standard die, they are integers from 1 to 6, and testing possible values reveals only two valid pairs: b = 2 when c = 1, and b = 4 when c = 4. The total number of possible outcomes for rolling two dice is 36, so the probability is the 2 successful outcomes divided by 36, which simplifies to 1/18.

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