Multiple choice

If $\displaystyle a, : b, : c : \in : R$ and $\displaystyle 3b^{2} - 8ac < 0$, then the equation $\displaystyle ax^{4} + bx^{3} + cx^{2} + 5x + 7 = 0$ has

  1. all real roots

  2. all are imaginary roots

  3. can not have all real roots

  4. exactly two real and two imaginary roots
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The condition 3b^2 - 8ac < 0 relates to the discriminant of the derivative or properties of the polynomial. For a quartic to have all real roots, the discriminant must be non-negative. This condition implies it cannot have all real roots.

AI explanation

If the quartic equation has four real roots r1, r2, r3, and r4, Vieta's formulas tell us that the sum of the products of the roots taken two at a time equals c/a. This sum is represented by (1/2)[(r1+r2+x3+r4)^2 - (r1^2+r2^2+r3^2+r4^2)], which clearly must be less than or equal to (1/2)(-b/a)^2. Multiplying by 8a^2 yields 4ac <= b^2, which directly contradicts the given condition that 3b^2 - 8ac < 0. Therefore, the equation cannot have all real roots.

Keep practicing — related questions