Multiple choice

Discuss the nature of the roots of the equation $\displaystyle 4ax^{2}+3bx+2c=0$, where $a, b, c> \displaystyle \in R$ and are connected by the relation $a+b+c=0$.

  1. All roots are imaginary.

  2. All roots are real.
  3. They can be real or imaginary depends on the value of $a,b,c$.

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The discriminant of the quadratic equation is D = 9b^2 - 32ac. Substituting b = -(a + c) into this expression yields D = 9(a + c)^2 - 32ac = 9a^2 - 14ac + 9c^2. Since the discriminant of this quadratic form in terms of a and c is negative, the expression 9a^2 - 14ac + 9c^2 is always non-negative for all real values of a and c, ensuring the roots are always real.

AI explanation

Evaluate the given quadratic 4ax^2 + 3bx + 2c = 0 at x = 1, which gives 4a + 3b + 2c. Using the given relation a + b + c = 0, we can substitute c = -a - b to get 4a + 3b + 2(-a - b) = 2a + b. This means 2a + b = 4a + 3b + 2c - 2(a + b + c). Since a + b + c = 0, evaluating at x = 1 is equivalent to evaluating 4a + 3b + 2c at x = 1, so x = 1 is a root. Because one real root exists for this quadratic equation with real coefficients, the other root must also be real. Thus, all roots are real.

Keep practicing — related questions