Multiple choice

If 2 is a root of the equation ${x}^{2}+bx+12=0$ and the equation ${x}^{2}+bx+q=0$ has equal roots, then q=

  1. 8

  2. -8

  3. 16

  4. -16
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C Correct answer
Explanation

Since 2 is a root of x^2 + bx + 12 = 0, 4 + 2b + 12 = 0, so b = -8. The second equation is x^2 - 8x + q = 0. For equal roots, the discriminant b^2 - 4ac must be 0. Thus, (-8)^2 - 4(1)(q) = 0, which means 64 = 4q, so q = 16.

AI explanation

Since 2 is a root of the equation x^2 + bx + 12 = 0, we substitute x = 2 to get 2^2 + 2b + 12 = 0. Solving this gives 4 + 2b + 12 = 0, which simplifies to 2b = -16 and b = -8. The second equation becomes x^2 - 8x + q = 0, and because it has equal roots, its discriminant must be zero: (-8)^2 - 4(1)(q) = 0. Solving 64 - 4q = 0 gives q = 16.

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