Multiple choice

If $a, b, c$ are real distinct numbers satisfying the condition $a + b + c = 0$, then the roots of the quadratic equation $3ax^2 + 5bx + 7c = 0$ are

  1. positive

  2. negative

  3. real and distinct

  4. imaginary
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The discriminant D = (5b)^2 - 4(3a)(7c) = 25b^2 - 84ac. Since a+b+c=0, b = -(a+c). Substituting b gives 25(a+c)^2 - 84ac = 25a^2 + 50ac + 25c^2 - 84ac = 25a^2 - 34ac + 25c^2. Since a and c are real and distinct, this discriminant is always positive, ensuring real and distinct roots.

AI explanation

Since a + b + c = 0, let a = -b - c. Substitute this into the quadratic 3ax^2 + 5bx + 7c = 0 to get 3(-b - c)x^2 + 5bx + 7c = 0, which rearranges to (x - 1)(-3bx - 7c) = 0, showing x = 1 is a root. If the roots were equal, the discriminant would be zero and (5b)^2 - 4(3a)(7c) would equal zero, which implies 25b^2 = 84ac; substituting a = -b - c yields 25b^2 + 84bc + 84c^2 = 0, but this is strictly positive for real b and c because its discriminant (84c)^2 - 4(25)(84c^2) = -4984c^2 is negative. Therefore, the quadratic equation has two real and distinct roots.

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