Multiple choice

The equation $ax^{2}-2bx+c=0, bx^{2}-2cx+a=0$ and $cx^{2}-2ax+b=0$ will have only positive roots provided

  1. $a,b,c$ are all positive
  2. $a,b,c$ are all negative
  3. $a>b>c$ or $a < b < c$
  4. $a=b=c$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For all three quadratic equations to have real roots, their discriminants must be non-negative, which gives b^2 >= ac, c^2 >= ab, and a^2 >= bc. Multiplying these inequalities yields a^2 * b^2 * c^2 >= a^2 * b^2 * c^2, which can only hold as an equality when a = b = c. Under this condition, the equations simplify to a(x - 1)^2 = 0, yielding the positive root x = 1.

AI explanation

For all three equations to have identical positive roots, they must be multiples of the same base equation. This requires the coefficients of the x^2 term, the x term, and the constant term to be proportional across all three equations, meaning a/b = b/c = c/a. This equality implies that a = b = c. The result is a=b=c.

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