Multiple choice

If one root of the equation x2 + mx + 12 = 0 is 4 and the equation x2 + mx + n = 0 has equal roots, then the value of n is

  1. 49/4

  2. 4/49

  3. 4

  4. ¼
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A Correct answer
Explanation

Since 4 is a root of x^2 + mx + 12 = 0, substitute x=4 to get 16 + 4m + 12 = 0, which gives 4m = -28, so m = -7. The second equation x^2 - 7x + n = 0 has equal roots, meaning its discriminant D = b^2 - 4ac = 0; thus (-7)^2 - 4(1)(n) = 0, which leads to 49 - 4n = 0, or n = 49/4.

AI explanation

Substitute the known root x = 4 into the first equation, x^2 + mx + 12 = 0, to get 16 + 4m + 12 = 0. Solving this gives 4m = -28, so m equals -7. For the second equation x^2 + mx + n = 0 to have equal roots, its discriminant must be zero. Setting m^2 - 4n = 0 and substituting m = -7 gives 49 - 4n = 0, which solves to n = 49/4.

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