Multiple choice

A real-valued function 𝑓(π‘₯)=π‘Žπ‘₯2 + 𝑏π‘₯ + 𝑐 (where π‘Ž β‰  0) is defined such that 5 is one of the roots of the equation 𝑓(π‘₯) = 0, and, 3𝑓(6) = - 2𝑓(3). Find the value of π‘Ž + 𝑏 + 𝑐.

    • 28
    • 20

  3. 6

  4. Cannot be determined
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

f(5) = 25a + 5b + c = 0. 3f(6) = 3(36a + 6b + c) = 108a + 18b + 3c. -2f(3) = -2(9a + 3b + c) = -18a - 6b - 2c. Setting them equal: 108a + 18b + 3c = -18a - 6b - 2c, so 126a + 24b + 5c = 0. We have two equations with three variables, so a+b+c cannot be uniquely determined.

AI explanation

If 5 is a root of f(x) = 0, we can express the quadratic as f(x) = a(x - 5)(x - k). Applying the given condition 3f(6) = -2f(3) leads to 3a(6 - 5)(6 - k) = -2a(3 - 5)(3 - k). Simplifying this yields 18 - 3k = 4k - 12, which solves to k = 30/7. This fixes the ratio of the coefficients, but since the leading coefficient a can be any non-zero number, infinitely many equations satisfy the conditions. The exact value of a + b + c, which equals f(1), cannot be determined.

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