Multiple choice

If $a,b,c\epsilon R,a\neq 0$ and $\left(b-1\right)^{2}<4ac$, then the number of real roots of the system of equation $\left(in\ three\ unknowns\ x_{1}, x_{2}, x_{3}\right)ax_{1}^{2}+bx_{1}+c=x_{2},ax_{2}^{2}+bx_{2}+c=x_{3},ax_{3}^{2}+bx_{3}+c=x_{1}$ is

  1. $0$
  2. $1$
  3. $2$
  4. $3$
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A Correct answer
Explanation

Rewrite each equation as x2 - x1 = ax1^2 + (b - 1)x1 + c, and similarly for the other two variables. The quadratic on the right has discriminant (b - 1)^2 - 4ac < 0, so it always has the same strict sign. This would force x2, x3, and x1 to change in one direction cyclically, which is impossible, so there are no real solutions.

AI explanation

By analyzing the cyclic system of equations, any real root x must satisfy the condition that (x1 - x2) times the sum of a specific expression equals zero. Given the condition (b-1) squared < 4ac, the discriminant of a related quadratic is negative, implying the expression inside cannot be zero. Therefore, the differences between the variables must be zero, leading to a contradiction under the given strict inequality, meaning the system has 0 real roots.

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