Multiple choice

Consider the quadratic equation $ax^{2} + bx + (a^{2} + b^{2} + c^{2} - ab - bc - ca) = 0$ (where $a,b, c$ are distinct real numbers) If the equation does not have any real roots then which of the following is always true.

  1. $b > 0$
  2. $b < 0$
  3. $a > 0$
  4. $a < 0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The discriminant D = b^2 - 4a(a^2 + b^2 + c^2 - ab - bc - ca). For no real roots, D < 0. D = b^2 - 4a^3 - 4ab^2 - 4ac^2 + 4a^2b + 4abc + 4a^2c. This expression is complex, but for a quadratic ax^2+bx+k=0 to have no real roots, if a > 0, the vertex must be above the x-axis.

AI explanation

The discriminant of the quadratic equation is D = b^2 - 4a(a^2 + b^2 + c^2 - ab - bc - ca). Factoring part of the constant term, we know that a^2 + b^2 + c^2 - ab - bc - ca = 1/2[(a - b)^2 + (b - c)^2 + (c - a)^2], which is strictly positive since a, b, c are distinct. Since the equation has no real roots, its discriminant must be less than zero, meaning a and the positive constant term must have the same sign. Therefore, the coefficient of x^2, which is a, must be strictly positive.

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