Multiple choice

Let $a, b, c\epsilon R_{0}$ and $1$ be a root of the equation $ax^{2} + bx + c = 0$, then the equation $4ax^{2} + 3bx + 2c = 0$ has

  1. Imaginary roots

  2. Real and equal roots

  3. Real and unequal roots

  4. Rational roots
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C Correct answer
Explanation

Since 1 is a root, a + b + c = 0, so c = -a - b. The discriminant of 4ax^2 + 3bx + 2c = 0 is 32a^2 + 32ab + 9b^2, which is positive for a nonzero. Therefore, the roots are real and unequal.

AI explanation

Because 1 is a root of ax^2 + bx + c = 0, we substitute x = 1 to get a + b + c = 0, meaning c = -(a + b). For the new equation 4ax^2 + 3bx + 2c = 0, we substitute c to find the discriminant is (3b)^2 - 4(4a)(2c) = 9b^2 - 32a(-a - b) = 9b^2 + 32a^2 + 32ab. This discriminant simplifies to (4a + b)^2 + 16a^2 + 8ab, which is positive, proving the roots are real and unequal.

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