Multiple choice

If the roots of the equation (a + b)x2 + (a + 2b + c)x + (b + c) = 0 are equal, which of the following statements is always true?

  1. a = c

  2. a + b + c = 0

  3. a = b = c

  4. b = a + c
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A Correct answer
Explanation

For the quadratic equation to have equal roots, the discriminant D = b'^2 - 4a'c' must be 0. Here a' = a+b, b' = a+2b+c, c' = b+c. (a+2b+c)^2 - 4(a+b)(b+c) = 0. Expanding this: (a+2b+c)^2 - 4(ab+ac+b^2+bc) = 0. a^2 + 4b^2 + c^2 + 4ab + 2ac + 4bc - 4ab - 4ac - 4b^2 - 4bc = 0. Simplifying: a^2 - 2ac + c^2 = 0, which is (a-c)^2 = 0. Therefore, a = c.

AI explanation

For the equation (a + b)x^2 + (a + 2b + c)x + (b + c) = 0 to have equal roots, its discriminant must be zero, meaning B^2 - 4AC = 0. Substituting the coefficients gives (a + 2b + c)^2 - 4(a + b)(b + c) = 0. Expanding both parts yields a^2 + 4b^2 + c^2 + 4ab + 4bc + 2ac - 4ab - 4ac - 4b^2 - 4bc = 0, which simplifies to a^2 - 2ac + c^2 = 0. This factors perfectly into (a - c)^2 = 0, proving that a = c.

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